FMM11

Friday Morning Math Problem

Inferring Rule from Sequence

             1 = 1
         1 - 4 = -(1 + 2)
     1 - 4 + 9 = 1 + 2 + 3
1 - 4 + 9 - 16 = -(1 + 2 + 3 + 4)

Guess the general law suggested by these examples, express it in suitable mathematical notation, and prove it.

Solution

The general law is: $1-4+9-16+...+(-1)^{n-1}n^{2}=(-1)^{n-1}{\frac {n(n+1)}{2}}$ $\sum _{k=1}^{n}(-1)^{k-1}k^{2}=(-1)^{n-1}{\frac {n(n+1)}{2}}$ This can be proved for the base case of n=1, in which case above equation reduces to 1=1 Then can show that if assume n is true, then n+1 holds $\sum _{k=1}^{n+1}(-1)^{k-1}k^{2}=(-1)^{n}{\frac {(n+1)(n+2)}{2}}$ If you split the left side into the sum from k=1 to n, (which can be reduced to a single term by using the identity above for the case of n, which you assume is true), and the last n+1 term written explicitly, you can do some algebraic manipulation to show it's equivalent to the right side.