## What Is A Triangular Number?

Contents

- 1 What is a triangular number in maths?
- 2 What is triangular numbers 1 to 100?
- 3 What is the 4th triangular number?
- 4 Why is 6 a triangular number?
- 5 What is 99th triangular number?
- 6 What is the 20th triangular number?
- 7 Is 666 a triangular number?
- 8 What is the 7th triangular number?
- 9 Is 6 8 and 14 a triangle?
- 10 Is there a 4 5 6 triangle?
- 11 What is the 25th triangular number?

## What is a triangular number in maths?

List Of Triangular Numbers – 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120,136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1225, 1275, 1326, 1378, 1431, and so on.

## What is triangular numbers 1 to 100?

There are 13 triangular numbers in the first 100 numbers. These are 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91.

## What is the 4th triangular number?

Solution – Many students will add new rows of counters, and make the 6th, 7th, 8th, 9th and 10th triangular numbers by construction. They will find that each new row requires one more counter than the previous one. This should lead them to see that the 10th triangular number is the 4th triangular number plus 5 + 6 + 7 + 8 + 9 + 10.

That is, 10 + 5 + 6 + 7 + 8 + 9 + 10. These can be added in order to give the 10th triangular number as 55. They may also see that the 1st triangular number has one on the bottom, the 2nd two on the bottom, the 3rd three and so on. The 4th triangular number is made up of 1 + 2 + 3 + 4 counters. So the result for the 10th triangular number can be written as 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10.

There is a quick way to add consecutive numbers like this (see ). When a string of numbers like this are added it is useful to ask yourself whether adding them in a different order makes the task more interesting. In this case, since the students will be familiar with ‘making ten’, it is natural for them to suggest adding (1+ 9) + (2 + 8) + (3 + 7) + (4 + 6) leaving only 5 and 10 to be added later.

- So, the 10th triangular number is 10 + 10 + 10 + 10 + 5 + 10.
- Another interesting way of adding the numbers is to add the first and the last, then the second and the second to last, and so on.
- This leads to (1+ 10) + (2 + 9) + (3+ 8) + (4 + 7) + (5 + 6).
- This simplifies to 11 + 11 + 11 + 11 + 11 = 55.
- Encourage the students to think like this when they work out the 20th triangular number.

So they have to add 1 + 2 + 3 + + 20 = (1 + 20) + (2 + 19) + + (10 + 11) = 10 x 21 = 210.

#### Why is 28 a triangular number?

Other properties – Triangular numbers correspond to the first-degree case of Faulhaber’s formula, Alternating triangular numbers (1, 6, 15, 28,,) are also hexagonal numbers. Every even perfect number is triangular (as well as hexagonal), given by the formula where M p is a Mersenne prime, No odd perfect numbers are known; hence, all known perfect numbers are triangular. For example, the third triangular number is (3 × 2 =) 6, the seventh is (7 × 4 =) 28, the 31st is (31 × 16 =) 496, and the 127th is (127 × 64 =) 8128.

- The final digit of a triangular number is 0, 1, 3, 5, 6, or 8, and thus such numbers never end in 2, 4, 7, or 9.
- A final 3 must be preceded by a 0 or 5; a final 8 must be preceded by a 2 or 7.
- In base 10, the digital root of a nonzero triangular number is always 1, 3, 6, or 9.
- Hence, every triangular number is either divisible by three or has a remainder of 1 when divided by 9: 0 = 9 × 0 1 = 9 × 0 + 1 3 = 9 × 0 + 3 6 = 9 × 0 + 6 10 = 9 × 1 + 1 15 = 9 × 1 + 6 21 = 9 × 2 + 3 28 = 9 × 3 + 1 36 = 9 × 4 45 = 9 × 5 55 = 9 × 6 + 1 66 = 9 × 7 + 3 78 = 9 × 8 + 6 91 = 9 × 10 + 1,

The digital root pattern for triangular numbers, repeating every nine terms, as shown above, is “1, 3, 6, 1, 6, 3, 1, 9, 9”. The converse of the statement above is, however, not always true. For example, the digital root of 12, which is not a triangular number, is 3 and divisible by three.

If x is a triangular number, then ax + b is also a triangular number, given a is an odd square and b = a − 1 / 8, Note that b will always be a triangular number, because 8 T n + 1 = (2 n + 1) 2, which yields all the odd squares are revealed by multiplying a triangular number by 8 and adding 1, and the process for b given a is an odd square is the inverse of this operation.

### Triangular Numbers Explained

The first several pairs of this form (not counting 1 x + 0 ) are: 9 x + 1, 25 x + 3, 49 x + 6, 81 x + 10, 121 x + 15, 169 x + 21,, etc. Given x is equal to T n, these formulas yield T 3 n + 1, T 5 n + 2, T 7 n + 3, T 9 n + 4, and so on. The sum of the reciprocals of all the nonzero triangular numbers is This can be shown by using the basic sum of a telescoping series : Two other formulas regarding triangular numbers are and both of which can easily be established either by looking at dot patterns (see above) or with some simple algebra. In 1796, Gauss discovered that every positive integer is representable as a sum of three triangular numbers (possibly including T 0 = 0), writing in his diary his famous words, ” ΕΥΡΗΚΑ! num = Δ + Δ + Δ “.

This theorem does not imply that the triangular numbers are different (as in the case of 20 = 10 + 10 + 0), nor that a solution with exactly three nonzero triangular numbers must exist. This is a special case of the Fermat polygonal number theorem, The largest triangular number of the form 2 k − 1 is 4095 (see Ramanujan–Nagell equation ).

Wacław Franciszek Sierpiński posed the question as to the existence of four distinct triangular numbers in geometric progression, It was conjectured by Polish mathematician Kazimierz Szymiczek to be impossible and was later proven by Fang and Chen in 2007.

## Why is 6 a triangular number?

Maths in a minute: Triangular numbers A triangular number is a number that can be represented by a pattern of dots arranged in an equilateral triangle with the same number of dots on each side. For example: The first triangular number is 1, the second is 3, the third is 6, the fourth 10, the fifth 15, and so on. You can see that each triangle comes from the one before by adding a row of dots on the bottom which has one more dot than the previous bottom row. This means that the triangular number is equal to

There’s also another way we can calculate the triangular number. Take two copies of the dot pattern representing the triangular number and arrange them so that they form a rectangular dot pattern. This rectangular pattern will have dots on the shorter side and dots on the longer side, which means that the rectangular pattern contains dots in total. And since the original triangular dot pattern constitutes exactly half of the rectangular pattern, we know that the triangular number is

Note that with this consideration we have proved the formula for the summation of natural numbers, namely

Triangular numbers have lots of interesting properties. For example, the sum of consecutive triangular numbers is a square number, You can see this by arranging the triangular dot patterns representing the and triangular numbers to form a square which has dots to a side: Alternatively, you can see this using the formulas for the consecutive triangular numbers and :

What is more, alternating triangular numbers (1, 6, 15,,) are also (numbers formed from a hexagonal dot pattern) and every even is a triangular number. Triangular numbers also come up in real life. For example, a network of computers in which every computer is connected to every other computer requires connections.

And if in sports you are playing a round robin tournament, in which each team plays each other team exactly once, then the number of matches you need for teams is These two results are equivalent to the handshake problem we have on Plus before. We would like to thank Zoheir Barka who sent us the first draft of this article.

We will publish a lovely article by Barka about triangular numbers soon. In the mean time, you can read Barka’s article about beautiful patterns in multiplication tables, : Maths in a minute: Triangular numbers

## What is 99th triangular number?

The 99th triangular number = (99 x 100)/2 = 4950.

### What is the pattern of 1 1 2 3 5 8?

The Fibonacci sequence is a series of numbers where a number is the addition of the last two numbers, starting with 0, and 1. The Fibonacci Sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55

#### What is the pattern rule for 3 6 10 15?

Sequences can be linear, quadratic or practical and based on real-life situations. Finding general rules for sequences helps find terms in sequences that would otherwise take a long time to work out.

- Test

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Number sequences are sets of numbers that follow a pattern or a rule, Each number in a sequence is called a term, There are some special sequences that you should recognise. The most important of these are:

- square numbers: 1, 4, 9, 16, 25, 36,, – the nth term is \(n^2\)
- cube numbers: 1, 8, 27, 64, 125,, – the nth term is \(n^3\)
- triangular numbers: 1, 3, 6, 10, 15,, (these numbers can be represented as a triangle of dots). The term to term rule for the triangle numbers is to add one more each time: 1 + 2 = 3, 3 + 3 = 6, 6 + 4 = 10 etc.
- Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13,, (in this sequence you start off with 1 and then to get each term you add the two terms that come before it)

A sequence which increases or decreases by the same amount each time is called a linear sequence e.g.

### Why is 28 the perfect number?

Celebrate The Math Holiday Of ‘Perfect Number Day’ This June 28th While it might seem that calling a number ‘perfect’ is subjective, it has a mathematical definition, that only a very few numbers can meet. Get to know them today. Judd Schorr / GeekDad Perfection is an unattainable quest for which we all strive.

But for a number, mathematically, being ‘perfect’ has a very specific definition that only a few select numbers can fulfill. A number is perfect if all of its factors, including 1 but excluding itself, perfectly add up to the number you began with.6, for example, is perfect, because its factors — 3, 2, and 1 — all sum up to 6.28 is perfect too: 14, 7, 4, 2, and 1 add up to 28.

But perfect numbers aren’t common at all. There are only two more, 496 and 8,128, below a million. Only 50 total perfect numbers are known, even with a dedicated worldwide effort to computationally discover more. Yet they have deep connections to some of the greatest mathematical questions of our time.

While, in celebration of the fact that τ = 2π, you simply can’t top a celebration of numbers that are truly perfect. Pi, or 3.14159., is the ratio of a circle’s circumference to its diameter. Tau, which is the, circumference-to-radius ratio, is twice as large. But although 6.28. might seem like it deserves a June 28th celebration, perfect numbers are far more worthy.

Public domain The calendar numbers of June 28th — 6 and 28 — have some very special properties that are worthy of a celebration. Unless you were born in the year 496, or are a time-traveler back from the year 8128, the only perfect numbers that will ever appear on your calendar are 6 and 28.

- If you can factor a number into all of its divisors, you can immediately add them all up and discover, for yourself, whether your number is perfect or not.
- For the first few numbers, this is a straightforward task, and you can see that most numbers aren’t perfect at all: they’re either abundant or deficient.

The first few countable numbers are mostly deficient, but 6 is a perfect number: the first and, easiest one to discover.E. Siegel If you add up all the positive factors of any number not including itself, you’ll get a number that’s either smaller than, greater than, or exactly equal to the original number.

If you add up all the factors excluding itself and get a number that’s less than the original one you started with, we call that number deficient, All prime numbers are maximally deficient, since its only factors are 1 and itself, and all powers of two (4, 8, 16, 32, etc.) are minimally deficient, with their sums falling just 1 shy of being perfect.

On the other hand, you might add up all the factors of a number excluding itself and get a number that’s greater than the original number; those numbers are abundant, You might look at the table above and think abundant numbers are rare, but 18, 20, 24, 30, 36 and many more are abundant; they’re quite common as you start looking at larger and larger numbers.

The factors of the first four perfect numbers. If you exclude the numbers themselves, all the other, factors (or divisors) sum up to the number in question, proving that they meet the criteria for perfect numbers.E. Siegel But perfect numbers — what Euclid called “τέλειος ἀριθμός” — are rare! For over a thousand years, only those first four were known.

You might look at these numbers, the ones that happen to be perfect, and start to notice a pattern here as to how these numbers can be broken down. They’re all the result of multiplying 2 to some power, let’s call it X, by a prime number. And interestingly, the prime number you’re multiplying it by is always equal to one less than double what 2 X is.

Different ways of breaking down the first four perfect numbers reveal a suggestive pattern as to how, they might be generated.E. Siegel There’s a good reason for this. Remember, all powers of two — numbers like 2, 4, 8, 16, 32, etc. — are minimally deficient, where they were just 1 shy of being perfect numbers.

At the same time, all prime numbers are maximally deficient, where their only factors are 1 and themselves. This means there are possible combinations of powers of two and prime numbers, minimally and maximally deficient numbers, that have a chance to be perfect themselves.

Not every minimally deficient and maximally deficient combination of numbers gives you a perfect number, though. If you look at the “prime factor breakdown” of perfect numbers, it looks like there’s a pattern for generating them! In fact, you might guess that the pattern goes something like this: The pattern that you might guess for all perfect numbers, based on the prime numbers we know, can,

only give you candidate perfect numbers. Many of these are not primes, and do not generate perfect numbers.E. Siegel After all, the first four prime numbers are 2, 3, 5 and 7, so you might think if we simply plugged prime numbers into this formula we stumbled into at the right — where n is a prime number and the formula is 2 ( n -1) * (2 n – 1) — we’d start generating perfect numbers.

- And you might think that this works for all primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, and so on.
- As it turns out, this is a great way to generate candidate perfect numbers, but not necessarily perfect numbers themselves.
- In fact, all known perfect numbers do follow this formula, where n is a prime number and “2 ( n -1) * (2 n – 1)” gives you a perfect number.

But it isn’t true that all prime numbers generate a perfect number; it only works for a select few! The first five perfect numbers, and some interesting numerical properties that they exhibit in terms, of generating them. Wikipedia page on Perfect Numbers The one you might think ought to have been the 5th perfect number — 2096128, which is 2 10 * ( 2 11 – 1 ) — is actually an abundant number.

It isn’t just random; there’s a reason. For 2, 3, 5, and 7, the (2 n – 1) part of the equation gave prime numbers: 3, 7, 31, and 127. The reason 2096128 isn’t a perfect number is because that part in parentheses, 2 11 – 1 (which is 2047), isn’t itself prime! 2047 can be factored: 23 * 89, and therefore it isn’t prime.

Because of this, the number 2096128, or 2 10 * ( 2 11 – 1 ), isn’t a perfect number, either! It isn’t enough to take your formula, 2 ( n -1) * (2 n – 1), for n being just a regular prime number; you need to ensure that the (2 n – 1) in your formula gives you a prime number as well.

- This type of prime — where n is prime and (2 n – 1) is also prime — is called a,
- Named after hundreds of years ago, there are (as of 2018) only 50 of them known in all existence.
- And they rise in size very quickly! The ways to generate the first 16 perfect numbers, and the Mersenne Primes that they correspond to.

Note how quickly these numbers rise, and also how recently they were discovered. Up until the 1950s, only 12 Mersenne primes were known. Screenshot from Wikipedia / Mersenne Primes The largest of the is, at present, 2 77,232,917 – 1, which has over 23 million digits in it written out! It’s uncertain that this is the 50th Mersenne prime because, although the first 42 Mersenne primes have been verified to be in order, there are large untested gaps of candidate Mersenne primes out there.

The perfect number that this corresponds to contains a whopping 46,498,849 digits, and would take about 16,000 printed pages to display. There is also, believe it or not, a search that the computer-savvy among you can participate in: the, including for finding new ones! Why would people care about primes like the Mersenne Primes? Chris Caldwell of the University of,

Tennessee-Martin has a FAQ that explains why. Chris Caldwell / UT-Martin If you wanted a little conjecture as to how to break the current record, here’s a fun piece of information you may want to consider. In addition to the numbers 3, 7, and 127 (the 1st, 2nd and 4th Mersenne primes), the number 170,141,183,460,469,231,731,687,303,715,884,105,727 is a Mersenne prime as well (the 12th), with 38 digits in it.

That means that in addition to 6, 28, and 8,128, the following number is absolutely perfect as well: 14,474,011,154,664,524,427,946,373,126,085,988,481,573,677,491,474,835,889,066,354,349,131,199,152,128. Many have conjectured that it’s very likely that (2^170,141,183,460,469,231,731,687,303,715,884,105,727 – 1) is a Mersenne prime, too, and would be one containing — are you ready — over 10 37 digits! Why do I believe that? Because of a little pattern, first noticed centuries ago: A fascinating pattern in Mersenne primes that was noted by Euler hundreds of years ago; it may lead,

us to the largest Mersenne Prime of all, and it may give us a way, if the pattern continues infinitely, to generate arbitrarily large Mersenne Primes.E. Siegel The first four numbers that follow this pattern are definitely Mersenne primes, but is the fifth? And more over, is this a valid way to generate an infinite number of Mersenne primes? The discovery of the first billion digit Mersenne prime — that is a Mersenne prime with only 10 9 (or more) digits — will net you a cool quarter-of-a-million dollars, but only if you can verify it! A more conceivable test, although it will only get you to around 6 × 10 8 digits (and a less lucrative ), would be to test whether (2^2,147,483,647 – 1) is a Mersenne prime.

- Leonhard Euler, famed mathematician, discovered the Mersenne Prime 2^31-1, which corresponds to a,
- Perfect number.
- Discovered in 1772 by Euler, it remained the largest known prime for over 90 years.
- There is an unproven conjecture that 2^2,147,483,647 – 1 is a Mersenne Prime, too.
- Jakob Emanuel Handmann, painter Many candidate Mersenne primes have been shot down by showing they can be factored, usually into two primes.

Just as 2047 = 23 * 89, many other candidate Mersenne primes have been shown not to be. In 1903, it was already known that ( 2 67 – 1) was not a Mersenne prime, but no one knew what its factors were. gave a talk to the American Mathematical Society entitled “On the Factorization of Large Numbers.” On the left side of the board, he computed ( 2 67 – 1), which he showed equaled 147,573,952,589,676,412,927.

On the right, he wrote 193,707,721 × 761,838,257,287, and spend his hour lecture saying nothing and working it out. At the end, when he showed both sides were equal, he sat down to a standing ovation, allegedly the first one ever given at a mathematics talk. Today, checking a possible factorization is much easier to do with a robust computer program like,

Mathematica than it was by hand many decades ago.E. Siegel / Mathematica The largest candidate Mersenne prime that’s been proven to be factorable so far is ( 2 1,168,183 – 1), which was shown (recently, in February 2014) to be able to be factored into 54,763,676,838,381,762,583 (which is prime) and a 351,639-digit number, which is thought to be prime as well.

It has been proven that all the even perfect numbers that exist are of the form that are generated by Mersenne primes that follow (2 n – 1), and it is conjectured (but not yet proven) that there are no odd perfect numbers; I have a feeling that accomplishing the latter (or, somehow, finding an odd perfect number) would be one of the greatest mathematical achievements of the century! Computer programs with enough computational power behind them can brute-force analyze a candidate,

Mersenne prime to see if it corresponds to a perfect number or not. For small numbers, this can be accomplished easily; for large numbers, this task is extremely difficult. C++ program originally from proganswer.com So that’s what a perfect number is, and a whole bunch of interesting math behind it.

### Is 72 a triangular number?

What is the Triangular Number? – The triangular number is a number that is represented by the sequence of numbers 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190. The triangular number is found by adding the numbers in the sequence together.

## What is the 20th triangular number?

How to Find Triangular Numbers – An equilateral triangle contains objects called triangular numbers (also referred to as triangle numbers). Triangular number N is equal to the sum of all “n” natural numbers from “1” to “n” and is the number of black dots in a triangular pattern with n black dots on each side.

An arrangement of triangular numbers, starting with the 0th triangular number, is as follows: Numbers 0 through 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, The following formulas can be used to calculate the triangle numbers: \ \ represents the Binomial coefficient.

It is written as “n plus one to one choose two” and it represents the number of distinct pairs that can be chosen from (n + 1) items. Imagine a “half-square” pattern of objects corresponding to each triangular number Tn. By reversing the pattern and creating a rectangular image, the number of objects doubles, giving a rectangle with dimensions n x (n+1), which is also the number of items in the rectangle.

#### Is a 3 4 5 triangle a thing?

A Pythagorean triple is a special case where the lengths of all the sides of a right triangle are whole numbers. The 3-4-5 triangle is the smallest and best known of the Pythagorean triples.

### How do you find the 25th triangular number?

Answer. In this case you want th 25th = 25*26/2 = 325.

## Is 666 a triangular number?

666 is the sum of the first thirty-six natural numbers, which makes it a triangular number :. Since 36 is also triangular, 666 is a doubly triangular number. Also, 36 = 15 + 21 where 15 and 21 are triangular as well, whose squares (15 2 = 225 and 21 2 = 441) add to 666 and have a difference of 216 = 6 × 6 × 6.

#### Why 9 is not a triangular number?

Answer: The correct answer is Option (b) 9. Step-by-step explanation: A triangular number is one that may be represented by a dot pattern with the same number of dots on each side of an equilateral triangle. The initial triangular number is 1, followed by triangular numbers 3, 6, 10, 15, and so on. a three-sided number is a fictitious number that may be represented as an equilateral triangular grid of elements with each row having one more element than the one before it. A number that may be expressed as the sum of the first n positive integers is referred to as a triangular number. so, 3 is a triangular number. is a triangular number., so 10 is also triangular number. The number which is not triangular number = 9. Find more like this here: brainly.in/question/54177468 #SPJ2

### Why is 36 a triangular number?

In mathematics – 36 depicted as a triangular number and as a square number 36 is both the square of six, and the eighth triangular number or sum of the first eight non-zero positive integers, which makes 36 the first non-trivial square triangular number, Aside from being the smallest square triangular number other than 1, it is also the only triangular number (other than 1) whose square root is also a triangular number.36 is also the eighth refactorable number, as it has exactly nine positive divisors, and 9 is one of them; in fact, it is the smallest number with exactly nine divisors, which leads 36 to be the 7th highly composite number, with exactly eight solutions ( 37, 57, 63, 74, 76, 108, 114, 126 ) to the Euler totient function Adding up some subsets of its divisors (e.g., 6, 12, and 18) gives 36; hence, it is also the eighth semiperfect number, This number is the sum of the cubes of the first three positive integers and also the product of the squares of the first three positive integers.36 is the number of degrees in the interior angle of each tip of a regular pentagram,

- The thirty-six officers problem is a mathematical puzzle with no solution,
- The number of possible outcomes (not summed) in the roll of two distinct dice,36 is the largest numeric base that some computer systems support because it exhausts the numerals, 0–9, and the letters, A-Z.
- See Base 36,
- The truncated cube and the truncated octahedron are Archimedean solids with 36 edges.

The number of domino tilings of a 4×4 checkerboard is 36. Since it is possible to find sequences of 36 consecutive integers such that each inner member shares a factor with either the first or the last member, 36 is an Erdős–Woods number, The sum of the integers from 1 to 36 is 666 (see number of the beast ).

## What is the 7th triangular number?

Asked 8 years, 2 months ago Viewed 2k times $\begingroup$ For example, the 7th triangular number is 28. How do I create something that will tell me that 28 is the 7th triangular number? This is what I have created. nthtri := Part, 2] & This is what happens when executed.

nthtri returns This newly defined function solves for the two roots of the quadratic equation. Since Mathematica lists the roots in increasing order, and since there are only two roots (one negative, one positive), by taking the 2nd root, you will find the “n” for a particular triangular number. I think my current method is unprofessional and inelegant.

How do I create a function that will return a singular output value? For example, I would like nthtri to return 7 Edit: As suggested below by @Guesswhoitis, using Root works. nthtri := Root & But I am still looking for more solution methods. I got lucky that solving for triangular numbers involves a quadratic equation that strictly has one positive and one negative root. MarcoB 65.3k 17 gold badges 90 silver badges 184 bronze badges asked Jun 29, 2015 at 12:31 user155812 user155812 505 3 silver badges 8 bronze badges $\endgroup$ 8 $\begingroup$ Following the comment and your own implementation with Root it is better to define this: nthtri := Root nthtri 7 This has no unlocalized Symbols as your own version had. With your code if you set x = 5 before using it it will fail. answered Jun 29, 2015 at 12:51 Mr.Wizard Mr.Wizard 268k 34 gold badges 576 silver badges 1346 bronze badges $\endgroup$ $\begingroup$ f is the function you are looking for: f := (1/2)*(-1 + Sqrt); t := (1/2)*n*(1 + n); Table], ] answered Jun 29, 2015 at 12:43 $\endgroup$ 3 $\begingroup$ Why not replace # with n and take second part as your function? (x /. Solve)] (*1/2 (-1 + Sqrt)*) f := 1/2 (-1 + Sqrt) f@28 (*7*) With}]] answered Jun 30, 2015 at 13:07 martin martin 8,542 4 gold badges 22 silver badges 67 bronze badges $\endgroup$ $\begingroup$ If $t, n$ are the triangular number and its position, then you want $n(n+1) = 2t$ Completing the square $n^2 + n + \frac = 2t + \frac $ and solving for positive n: $n = \sqrt } – \frac $ nthtri := Sqrt – 0.5; answered Jun 29, 2015 at 13:56 $\endgroup$ 1

## Is 6 8 and 14 a triangle?

You can create a triangle using side lengths 6, 8, and 14.

## Is there a 4 5 6 triangle?

The three numbers 4, 5, 6 make a Pythagorean Triple (they could be the sides of a right triangle).

## What is the 25th triangular number?

Triangular number – Simple English Wikipedia, the free encyclopedia The first six triangular numbers A triangular number is a number that is the of all of the up to a certain number. When formed using regularly spaced dots, they tend to form a shape of, hence the name.

- For example, 10 is a “triangular number” because 1 + 2 + 3 + 4 = 10,
- The first 25 triangular numbers are: 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, and so on.
- A triangular number is calculated by the equation: n ( n + 1 ) 2 }},

### Is 72 a triangular number?

What is the Triangular Number? – The triangular number is a number that is represented by the sequence of numbers 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190. The triangular number is found by adding the numbers in the sequence together.

#### What is the triangular number 55?

List of Triangular Numbers – The following is the triangular numbers list to 10000 that might come handy: 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120,136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1225, 1275, 1326, 1378, 1431, 1485, 1540, 1596, 1653, 1711, 1770, 1830 etc.

#### What is the pattern rule for 3 6 10 15?

Sequences can be linear, quadratic or practical and based on real-life situations. Finding general rules for sequences helps find terms in sequences that would otherwise take a long time to work out.

- Test

- 1
- 2
- 3
- 4
- Page 1 of 4

Number sequences are sets of numbers that follow a pattern or a rule, Each number in a sequence is called a term, There are some special sequences that you should recognise. The most important of these are:

- square numbers: 1, 4, 9, 16, 25, 36,, – the nth term is \(n^2\)
- cube numbers: 1, 8, 27, 64, 125,, – the nth term is \(n^3\)
- triangular numbers: 1, 3, 6, 10, 15,, (these numbers can be represented as a triangle of dots). The term to term rule for the triangle numbers is to add one more each time: 1 + 2 = 3, 3 + 3 = 6, 6 + 4 = 10 etc.
- Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13,, (in this sequence you start off with 1 and then to get each term you add the two terms that come before it)

A sequence which increases or decreases by the same amount each time is called a linear sequence e.g.