- #1

simmonj7

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## Homework Statement

For n ∈

**N**

**(the natural numbers), ∑ (-1)**

For proof by induction, to show that the statement, P(n) is true for all n ∈^{i}i^{2}(sum from i=1 to n) = (-1)^{n}n(n+1)/2.## Homework Equations

For proof by induction, to show that the statement, P(n) is true for all n ∈

**N****you must show that P(1) is true and P(k+1) is true whenever P(k) is true.**

Ok so here is where I am at:

We will use induction on n.

Basis step:

For n = 1, we have ∑ (-1)

Inductive step:

Suppose ∑ (-1)

And here is where I am stuck. Cause right now I have the second part i.e. (k+1)((k+1)+1)/2 right, however I can't see how to get the exponents to become -1

Thanks## The Attempt at a Solution

Ok so here is where I am at:

We will use induction on n.

Basis step:

For n = 1, we have ∑ (-1)

^{1}1^{2}(sum from i=1 to 1) = (-1)^{1}1^{2}= -1 and (-1)^{1}1(1+1)/2 = -1. So, when n =1, ∑ (-1)^{i}i^{2}(sum from i=1 to n) = (-1)^{n}n(n+1)/2.Inductive step:

Suppose ∑ (-1)

^{i}i^{2}(sum from i=1 to k) = (-1)^{k}k(k+1)/2. Then ∑ (-1)^{i}i^{2}(sum from i=1 to k+1) = ∑ (-1)^{i}i^{2}(sum from i=1 to k) + (-1)^{k+1}(k+1) = (-1)^{k}k(k+1)/2 + (-1)^{k+1}(k+1) by the inductive hypothesis. So, -1^{k}k(k+1)/2 + 2(-1)^{k+1}(k+1)/2 = (-1^{k}k(k+1) + 2(-1^{k+1})(k+1))/2 = ((-1^{k}+ -1^{k+1})(k+1)(k+1)+1)) /2.And here is where I am stuck. Cause right now I have the second part i.e. (k+1)((k+1)+1)/2 right, however I can't see how to get the exponents to become -1

^{k+1}Thanks