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Proof by induction? No Idea what I should do :(

FriendlyCashew

New member
Jul 13, 2020
2
find the general rule and prove by induction

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

I created this so far, but don't know if I am even going the correct direction

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Attachments

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,694
find the general rule and prove by induction

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

I created this so far, but don't know if I am even going the correct direction

View attachment 10459
You have the correct formula, except that $n^n$ should be $n^2$. I think your next step should be to simplify the right side of the formula by using the formula (which you probably know?) for the sum of the numbers $1$ to $n$. After that, you should be able to apply the method of induction.
 

topsquark

Well-known member
MHB Math Helper
Aug 30, 2012
1,121
find the general rule and prove by induction

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

I created this so far, but don't know if I am even going the correct direction

View attachment 10459
Actually, I think what you want is
\(\displaystyle \sum_{n = 1}^k (-1)^{n - 1} n^2 = (-1)^{k + 1} \sum_{n = 1}^k n\)

Does this work for some value of k? (Sure, try k = 1.) So if it works for k, does it work for k + 1?

-Dan
 

FriendlyCashew

New member
Jul 13, 2020
2
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1594726804881.png

Is this correct?
 

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,694
find the general rule and prove by induction

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

I created this so far, but don't know if I am even going the correct direction

View attachment 10459
The equation that you are trying to prove by induction is \(\displaystyle \sum_{n = 1}^k (-1)^{n - 1} n^2 = (-1)^{k + 1} \sum_{n = 1}^k n\). Your first attempt at this was wrong, and so was mine (sorry about that). But topsquark got it right. The next step is to use the fact that \(\displaystyle \sum_{n = 1}^k n = \frac12k(k+1)\). So you want to prove that \(\displaystyle \sum_{n = 1}^k (-1)^{n - 1} n^2 = \frac12(-1)^{k + 1}k(k+1)\). That equation is true when $k=1$ because both sides are then equal to $1$.

Now suppose that the equation is true for $k$. You want to show that it is also true for $k+1$, namely that \(\displaystyle \sum_{n = 1}^{k+1} (-1)^{n - 1} n^2 = \frac12(-1)^{k + 2}(k+1)(k+2)\). On the left side of the equation, that differs from the previous equation just by the addition of one extra term $(-1)^k(k+1)^2$. So starting with the known equation \(\displaystyle \sum_{n = 1}^k (-1)^{n - 1} n^2 = \frac12(-1)^{k + 1}k(k+1)\), add $(-1)^k(k+1)^2$ to each side to get \(\displaystyle \sum_{n = 1}^{k+1} (-1)^{n - 1} n^2 = \frac12(-1)^{k + 1}k(k+1) + (-1)^k(k+1)^2\).

Now can you simplify the right side of that equation to complete the proof?