Sum of Squares of n Natural Numbers
2026-02-28 23:13 Diff
https://brightchamps.com/en-us/math/numbers/sum-of-squares-of-n-natural-numbers

Here, in this section, we will learn how to derive the formula to find the sum of squares like 12 + 22 + 32 + … 

The formula used is 12 + 22 + 32 + … + n2 =  n (n+1) (2n +1) / 6 

To see how the formula is derived, we use of the process called mathematical induction. It makes sure that the formula works for one number, proving that it applies to other numbers as well. Follow the steps given below:-

Step 1: Check if the formula works for number 1. Therefore, we can take the value of ‘n’ as 1. 

LHS is 12 = 1

RHS = n (n+1) (2n +1) / 6 

Now substituting the value of ‘n’ as 1 in the formula, we get 1 (1+1) ((2 x 1) +1) / 6 =
1 x 2 x 36 =  6/6 = 1

Since both the LHS and RHS are the same, we can say that the formula works for n = 1

Step 2: Now let’s check if the formula works for the number ‘k’.
Assume that,

12 + 22 + 33 + … + k2 = k(k+1) (2k + 1) / 6

Since we are assuming for number ‘k’ to be true, this process is called the inductive hypothesis.
Now we have to check if it works for the number ‘k + 1’

Step 3: Let’s prove that the formula works for ‘k + 1’
12 + 22 + 33 + … + (k + 1)2 =  (k+1) (k+2) (2k+3) / 6

The sum of squares up to k is 12 + 22 + 33 + … + k2 = k(k+1) (2k + 1) / 6

Now we can add (k + 1)2 to both sides:-

12 + 22 + 32 + … + k2 + (k + 1)2 = k(k+1) (2k + 1) / 6 + (k + 1)2

Step 4: Take out the common factor (k + 1) from both side
k+1/ 6 [k (2k + 1) + 6 (k +1)]
By simplifying the brackets, we get:-

k+1/ 6 (2k2 + k + 6k + 6) = k+1/ 6 (2k2 + 7k + 6)

Step 5: Now factor (k+2) from (2k2 + 7k + 6) by splitting them
By doing so, we get:-
k+1 6 [2k (k + 2)  + 3 (k + 2)]
Factoring out (k + 2), we get:-

k+1 6 (k + 2) (2k + 3)

Step 6: We get the final answer as:- (k+1) (k + 2) (2k + 3) / 6

Since the formula is true for n=1, and we've shown that if it works for any number k, it also works for k+1, we can conclude that the formula is true for all natural numbers. Therefore, the sum of squares formula is:

S = 12 + 22 + 32 + 42 + …. + n2 = n (n+1) (2n +1) / 6