Sum of Squares

How to solve the given summation by using the sum of squares (k2) formula: formula, 1 example, and its solution.

Formula

Formula

The sum of k2, as k goes from 1 to n, is
n(n + 1)(2n + 1)/6.

This means
12 + 22 + 32 + ... + n2 = n(n + 1)(2n + 1)/6.

Example

Example

Solution

k(3k + 1) = 3k2 + k

The coefficient 3 gets out from the sigma.
k2 is in the sigma.

k is in the sigma.

Summation: How to Solve

Write the coefficient 3.

The sum of k2 is
n(n + 1)(2n + 1)/6.

The sum of k is
n(n + 1)/2.

Sum of k

So 3⋅n(n + 1)(2n + 1)/6 + n(n + 1)/2.

Cancel the denominator 3
and reduce the numerator 6 to, 6/3, 2.

The common factor of
n(n + 1)(2n + 1)/2 and +n(n + 1)/2
is n(n + 1)/2.

So write
n(n + 1)/2.

Write, n(n + 1)(2n + 1)/2 ÷ n(n + 1)/2,
(2n + 1.

And write, +n(n + 1)/2 ÷ n(n + 1)/2,
+1).

2n + 1 + 1 = 2n + 2

(2n + 2)/2 = (n + 1)

(n + 1)(n + 1) = (n + 1)2

So n(n + 1)2 is the answer.