1+3+6+10+...+n(n+1)/2=n(n+1)(n+2)/6 Brainly

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Jun 15, 2024

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Derivation of the Formula: 1+3+6+10+...+n(n+1)/2=n(n+1)(n+2)/6

Introduction

In mathematics, the formula 1+3+6+10+...+n(n+1)/2=n(n+1)(n+2)/6 is a well-known identity that represents the sum of consecutive triangular numbers. In this article, we will derive this formula and explore its significance in mathematics.

The Concept of Triangular Numbers

Triangular numbers are a sequence of numbers in which each number is the sum of the previous number and the next positive integer. The sequence of triangular numbers is: 1, 3, 6, 10, 15, ... . The nth triangular number can be expressed as n(n+1)/2.

Derivation of the Formula

To derive the formula, we can start by writing the sum of consecutive triangular numbers as:

1 + 3 + 6 + ... + n(n+1)/2

We can rewrite this sum as:

Σ(k=1 to n) [k(k+1)/2]

Using the properties of summation, we can rewrite the sum as:

(Σk=1 to n) [k(k+1)] / 2

Expanding the numerator, we get:

(Σk=1 to n) [k^2 + k] / 2

Splitting the sum into two separate sums, we get:

(Σk=1 to n) [k^2] / 2 + (Σk=1 to n) [k] / 2

Using the formulas for the sum of squares and the sum of an arithmetic series, we get:

[n(n+1)(2n+1)/6] / 2 + [n(n+1)/2] / 2

Simplifying the expression, we get:

n(n+1)(n+2)/6

Thus, we have derived the formula:

1+3+6+10+...+n(n+1)/2=n(n+1)(n+2)/6

Conclusion

In this article, we have derived the formula 1+3+6+10+...+n(n+1)/2=n(n+1)(n+2)/6, which represents the sum of consecutive triangular numbers. This formula has many applications in mathematics, particularly in number theory and combinatorics.