Question

Question:

1²+3²+5²+7²+...upto n terms.​

Answer 1

hello,,

ur answer...↓↓

Let 2k + 1 ne the general term of the sum. If we'll replace k by values from 0 to n, we'll get each odd natural number.

We'll raise to square and we'll expand the binomial:

⇒(2k+1)^2 = 4k^2 + 4k + 1

We'll take the summation:

⇒1^2 + 2^2 +...n^2 = Sum (2k+1)^2 = Sum 4k^2 + Sum 4k + Sum 1

⇒Sum (2k+1)^2 = 4*n*(n+1)*(2n+1)/6 + 4*n(n+1)/2 + n

⇒Sum (2k+1)^2 = [4*n*(n+1)*(2n+1) + 12n*(n+1) + 6n]/6

We'll remove the brackets:

⇒Sum (2k+1)^2 = [4n(n^2 + 3n + 1) + 12n^2 + 12n + 6n]/6

⇒Sum (2k+1)^2 = (4n^3 + 12n^2 + 4n + 12n^2 + 12n + 6n)/6

We'll combine like terms:

⇒Sum (2k+1)^2 = (4n^3 + 24n^2 + 22n)/6

⇒Sum (2k+1)^2 = 2n(2n^2 + 12n + 11)/6

⇒Sum (2k+1)^2 = n(2n^2 + 12n + 11)/3

The requested sum of the squares of the odd natural numbers is:

↓↓↓↓↓↓↓↓↓↓

1^2 + 2^2 + ...n^2 = Sum (2k+1)^2 = n(2n^2 + 12n + 11)/3\

regards @lava90 ( official lolwa queen )

Answer 2

PLEASE REFER THE ATTACHMENT...........!!