1 3 6 10 Formula

The triangular number sequence is the representation of the numbers in the course of equilateral triangle arranged in a series or sequence. These numbers are in a sequence of 1, three, six, 10, 15, 21, 28, 36, 45, and so on. The numbers in the triangular blueprint are represented by dots. The sum of the previous number and the order of succeeding number results in the sequence of triangular numbers. Nosotros will learn more here in this article.

• Triangles
• Sequences And Series Class xi
• Of import Questions Grade eleven Maths Chapter nine Sequences Series

What is Triangular Number?

A triangular number T_n is a figurative number that can exist represented in the form of an equilateral triangular filigree of elements such that every subsequent row contains an element more than the previous one.

List Of Triangular Numbers

0, one, iii, half-dozen, x, 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.

Sum of Triangular Numbers

In the blueprint of triangular numbers you will encounter, the next number in the sequence is added with an extra row. Let us explain in particular.

• Showtime number is 1
• In number ii,  a row is added with two dots to the first number
• In number iii, a row is added with three dots to the second number
• Again, in number 4, a row is added with four dots to the third number and so on

So the sequence formed here is in the pattern:

i, 1 + 2, i + two + 3, ane + 2 + 3 + 4,  and so on.

Formula For Triangular Number Sequence

Triangular numbers stand for to the get-go-degree example of Faulhaber's formula.

\(\begin{assortment}{l}T_{n}=\sum_{k=1}^{north}k=1+two+iii+…+northward=\frac{north(due north+1)}{2}\stop{assortment} \)

Where,

(n+i)/2 is the binomial coefficient. It represents the number of distinct pairs that can exist selected from N+one objects.

Further, (north+i)/ii can be expressed as:

\(\begin{assortment}{l}\frac{(n+1)!}{(north+i-2)!2!}\end{array} \)

This tin be further simplified as [n(n+1)]/two.

By the above formula, nosotros can say that the sum of n natural numbers results in a triangular number, or we can too say that continued summation of natural numbers results in a triangular number. The sum of ii consecutive natural numbers always results in a foursquare number.

\(\begin{assortment}{50}T1+T2= one+3=4=2^{2}\end{array} \)

and

\(\brainstorm{array}{l}T2+T3= iii+6=nine=3^{2}\end{array} \)

All even perfect numbers are triangular numbers, and every alternate triangular number is a hexagonal number given past the formula:

\(\begin{array}{l}M_{P}2^{p-1}=\frac{M_{p}(M_{p}+1)}{two}=T_{Mp}\end{array} \)

Where M_P is a Mersenne prime number.

For example, the third triangular number is (three × 2 =) six, the seventh is (7 × four =) 28, the 31st is (31 × 16 =) 496, and the 127th is (127 × 64 =) 8128.

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1 3 6 10 Formula,

Source: https://byjus.com/maths/triangular-numbers/

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