finding general law given a few statements

Question

I am doing a problem on induction and the first step is to guess the general law of the following statements:

$1=1$

$1-4=-(1+2)$

$1-4+9=1+2+3$

For the LHS i got that we are summing

$(-1)^{r+1}*r^2$ from $r=1$ to $n$ but i am quite stuck on the RHS.

maybe some formula including $(-1)^{n+1}$ i am not sure, how do you approach this kind of thing?

Thanks

Answer 1

Your statement should be

$$\sum_{i=1}^n(-1)^{i+1}i^2 = (-1)^{n+1}\sum_{i=1}^n i.$$

Answer 2

RHS: the $n$-th term is given by $\sum_{k = 1}^{n} (-1)^{k + 1} k^2$.

LHS: the $n$-th term ist given by $(-1)^{n + 1} \sum_{k = 1}^{n} k$.

You correctly spotted summing of the numbers or their squares, and the $(-1)^z$-term is placed outside the sum if the sign of all numbers changes with each new term and inside the sum if each summand has a different sign.