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 | High School Teacher | (Level 2) Educator

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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 

`1^2-2^2+3^2-4^2+ldots-(2k)^2+(2k+1)^2-(2k+2)^2=-(k+1)(2k+3)`

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

`LS=1^2-2^2+3^2-4^2+ldots-(2k)^2+(2k+1)^2-(2k+2)^2`

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

`=-2k^2-k+4k^2+4k+1-4k^2-8k-4`

`=-2k^2-5k-3`

`=-(k+1)(2k+3)`

`=RS`

By induction the result has now been proven.

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