Sum of Squares

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There are two types of ‘sum of squares’ formulas, where one of them is the sum of squares of two numbers and the other is the sum of squares of ‘n’ given values. The formula for sum of squares of two numbers ‘a’ and ‘b’ is given as a2 + b2 = (a + b)2 – 2ab. The formula for the sum of squares of ‘n’ values such as (12 + 22 + 32 + 42 + … + n2) = [n (n+1) (2n + 1)]/ 6. According to the given situation, either of the formulas is used for solving.

Example 1: Find the value of the sum of the squares of ‘1’ and ‘4’ written as, 12 + 42.

According to the sum of the squares formula, a2 + b2 = (a + b)2 – 2ab.

Given 12 + 42, which implies a = 1 and b = 4.

Therefore according to the sum of the squares formula, we get: 12 + 42 = (1 + 4)2 – 2*1*4

This gives: 12 + 42 = (5)2 – 8 ==> 12 + 42 = 25 – 8 = 17.

Hence the value of 12 + 42 = 17.
 
Example 2: Find the value of the sum of squares from ‘1’ to ‘5’.

Sum of the squares from ‘1’ to ‘5’ can be written as: 12+ 22+ 32+ 42+ 52.

Sum of squares of numbers from ‘1’ to ‘n’ written as:

(12+ 22+32+ … + n2) = [n (n+1) (2n + 1)]/ 6

Here n= 5.

So, 12+ 22+ 32+ 42+ 52 = [5 (5 + 1) (10 + 1)]/ 6.

This gives: (5* 6* 11)/ 6= 330/6 = 55.

Therefore the sum of squares from ‘1’ to ‘5’ is 55.
 

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