Question

the series
Find the sum to n terms .
1² +2²+ 3² +4² + ... + n^2​

Answer 1

Answer:

If n is even, the sum is -n*(n+1)/2.

That is because, 1^2 - 2^2 = -1*(1+2) [ Since a^2 - b^2 = (a-b)*(a+b)

Also, 3^2 - 4^2 = -1*(3+4)

Thus , in the sum , taking -1 common, we have,

-1* (1+2+3+4+ .. + n) = -n*(n+1)/2

If the value of n is odd, the sum is clearly ,

S(n-1) + n^2

= -n*(n-1)/2 + n^2

= -n^2/2 + n^2 + n/2

= n^2/2 + n/2

=n*(n+1)/2

<EOM>