Prove that: `1^2-2^2+3^2-4^2+ * * * +n^2=(-1)^(n-1)(n(n+1))/2`  Example 1^2 - 2^2 + .....+11^2=11(11+1)/2=66   How can I prove this mathematically. Thanks

Expert Answers
embizze eNotes educator| Certified Educator

Lets look at some terms of the sequence:





Thus the sequence is 1,-3,6,-10,15,-21,28,-36,...

We can use mathematical induction:

(1) Base case : n=1 `1^2=1=(1(2))/2` is true.

(2) For the inductive hypothesis:


(3) Assume for some `k>=1,kin NN` (k a natural number) that the inductive hypothesis is true.

``(a)If k is odd (so (k+1) is even) we need to show: `1^2-2^2+3^2-4^2+...+k^2-(k+1)^2=-((k+1)(k+2))/2`

We work on the left hand side:

By the inductive hypothesis `1^2-2^2+...+k^2=(k(k+1))/2` ; substituting we get:



`=(k^2+k-2k^2-4k-2)/2`   (Get a common denominator and add)


`=-((k+1)(k+2))/2` as required.

(b) If k is even, we need to show:


Again we work onthe left hand side; by the inductive hypothesis we have:

`1^2-2^2+3^2-...-k^2=-(k(k+1))/2` . Substituting we get:





`=((k+1)(k+2))/2` as required.