Use mathematical induction to show that :   1^2-2^2+3^2-4^2 ....- (2n)^2 = -n(2n+1)    I tried  but I  failed  so please help 

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lfryerda eNotes educator| Certified Educator

To use induction you need to prove that the statement is true for initial value(s) of n, and then assume ti is true for n=k to prove that it is true for n=k+1.

In this case, we see that the statement when n=1 is true since `LS=1^2-2^2=-3` and `RS=-1(3)=-3` .

Assume that `1^2-2^2+3^2-4^2+ldots-(2k)^2=-k(2k+1)` is true for some `n=k` .

Now let `n=k+1`.

Then we need to show that 


Start with the left side and work to the right side.


`=-k(2k+1)+(2k+1)^2-(2k+2)^2`   using the induction assumption





By induction the result has now been proven.