Question

if one square + 2 square + 3 square to + 10 square is equal to 385 so what is the value of 2 square + 6 square to 20 square​

Answer 1

Answer:We know sum squares of first n natural numbers is \frac{n(n+1)(2n+1)}{6}.

How to compute sum of squares of first n even natural numbers?

We need to compute 22 + 42 + 62 + …. + (2n)2

EvenSum = 22 + 42 + 62 + .... + (2n)2

       = 4 x (12 + 22 + 32 + .... + (n)2)

       = 4n(n+1)(2n+1)/6

       = 2n(n+1)(2n+1)/3

Example:

Sum of squares of first 3 even numbers =

                2n(n+1)(2n+1)/3

              = 2*3(3+1)(2*3+1)/3

              = 56

22 + 42 + 62 = 4 + 16 + 36 = 56

How to compute sum of squares of first n odd natural numbers?

We need to compute 12 + 32 + 52 + …. + (2n-1)2

OddSum  = (Sum of Squares of all 2n numbers) -

         (Sum of squares of first n even numbers)

       = 2n*(2n+1)*(2*2n + 1)/6 - 2n(n+1)(2n+1)/3

       = 2n(2n+1)/6 [4n+1 - 2(n+1)]

       = n(2n+1)/3 * (2n-1)

       = n(2n+1)(2n-1)/3

Example:

Sum of squares of first 3 odd numbers = n(2n+1)(2n-1)/3

                                     = 3(2*3+1)(2*3-1)/3

                                     = 35

12 + 32 + 52 = 1 + 9 + 25 = 35

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Answer 2

Question- Given that one square + 2 square + 3 square to + 10 square is equal to 385 then the value of 2 square + 4 square + 6 square to 20 square​ is equal to-

A. 770  B. 1540  C. 1155  D. $385^{2}$

The required value is 1540. (Option B)

Given:

$1^{2} +2^{2} +3^{2} .....+10^{2} =385$

To find:

The value of $2^{2} +4^{2} +6^{2}+ ... +20^{2}$

Solution:

Let the required sum be a.

So, $2^{2} +4^{2} +6^{2}+ ... +20^{2}$ =a

We see that the square of each number in the second series is obtained by taking 4 times the square of each number in the first series.

So, the required sum can be obtained by calculating 4 times the sum of squares of terms in the first series.

$2^{2} +4^{2} +6^{2}+ ... +20^{2}$ = 4×( $1^{2} +2^{2} +3^{2} .....+10^{2}$)

a=4 times 385

a=4×385

a=1540

$2^{2} +4^{2} +6^{2}+ ... +20^{2}$ =1540.

Therefore, the required value is 1540.

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