I would like to prove the following equation using induction. However that seems somehow impossible at least for me:
$\sum\limits_{k=1}^{2n} {(-1)^k \cdot k^2}=(2n+1)\cdot n$
I tried to show that some property $E()$ holds for $n=1$, $E(1)$ but I just get something which doesnt makes sense:
$(-1)^1 \cdot 1^2=(2 \cdot 1+1) \cdot 1$
$-1=3$
When $n=1$, the sum on the left hand side is from $k=1$ to $k=2$, which is
$$(-1)^1 \cdot 1^2 + (-1)^2 \cdot 2^2 = (-1) + 4 = 3$$