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Basically I'm trying to get this expression:

$${(-1)^nn(n+1)\over2} + (-1)^{(n+1)}(n+1)^2$$

In this form:

$$(-1)^{(n+1)}(n+1)(n+2)\over2$$

This is for a proof using mathematical induction, and I'm 99% sure that they DO equal each other. For some reason I'm having trouble doing this. What I've tried is setting 2 as the common denominator, and then factoring out a $(n+1)$ term and a $(-1)^{(n+1)}$ term.

After doing these steps, I ended up with:

$$(-1)^{(n+1)}(n+1)(2n+1)\over2$$

I'm not sure if I'm making a simple error, missing something, or if these two expressions are not equal to each other (but I seriously think that they are, because of the nature of the assignment that this question is from). Sometimes the simplest solutions are so hard to find.

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1 Answer 1

up vote 3 down vote accepted

Try this: $$ \frac{(-1)^{n}n(n+1)}{2}+(-1)^{n+1}(n+1)^2 = (-1)^n\bigg(\frac{n(n+1)}{2}-\frac{2(n+1)^2}{2}\bigg)\\ =\frac{(-1)^n(n+1)}{2}(n-2(n+1))\\ =\frac{(-1)^{n+1}(n+1)(n+2)}{2} $$

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Thanks! That answers my question. –  mh234 Oct 9 '12 at 2:29

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