FMM11: Difference between revisions
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(Created page with "{{FMM |title=Inferring Rule from Sequence |problem= <pre> 1 = 1 1 - 4 = -(1 + 2) 1 - 4 + 9 = 1 + 2 + 3 1 - 4 + 9 - 16 = -(1 + 2 + 3 + 4) </pre> Gu...") |
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The general law is: | The general law is: | ||
1 - 4 + 9 - 16 + ... + (-1)^ | <math> | ||
1 - 4 + 9 - 16 + ... + (-1)^{n-1} n^2 = (-1)^{n-1} \frac{n(n+1)}{2} | |||
(-1)^ | </math> | ||
<math> | <math> | ||
\sum_{k=1}^n (-1)^{k-1} k^2 = (-1)^{n-1} \frac{ | \sum_{k=1}^n (-1)^{k-1} k^2 = (-1)^{n-1} \frac{n(n+1)}{2} | ||
</math> | </math> | ||
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<math> | <math> | ||
\sum_{k=1}^{n+1} (-1)^{k-1} k^2 = (-1)^{n} \frac{ | \sum_{k=1}^{n+1} (-1)^{k-1} k^2 = (-1)^{n} \frac{(n+1)(n+2)}{2} | ||
</math> | </math> | ||
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. | 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. | ||
}} | }} | ||
Latest revision as of 22:36, 16 November 2019
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. |
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