Prove by induction
- Thread starter hikary3
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Let n=1? Then check we get 1 = 1. And now I don't know what next
lev888
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Have you done a proof by induction before? Have you followed a few examples? Please look up some videos on youtube, etc. Then come back and continue to step 2.
pka
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Next suppose that \(J\in\mathbb{Z}^+~\&~J>1\) is such that \(\displaystyle\sum\limits_{k = 1}^J {{k^3}} = {\left[ {\frac{{J(J + 1)}}{2}} \right]^2}\).
Then show that the sum works for \(J+1\) based on the above.
HallsofIvy
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To prove statement P(n), that depends upon the positive integer, n, "by induction" you need to
1) Prove P(1) is true.
2) Prove that if P(k) is true, then P(k+1) is also true.
One reason why so many "proofs by induction" involve sums is that the "n+1" case is just the "n" case with one more term!
Here \(\displaystyle \sum_{i=1}^{k+1} i^3= \sum_{i=1}^k i^3+ (k+1)^3\).
1) Prove P(1) is true.
2) Prove that if P(k) is true, then P(k+1) is also true.
One reason why so many "proofs by induction" involve sums is that the "n+1" case is just the "n" case with one more term!
Here \(\displaystyle \sum_{i=1}^{k+1} i^3= \sum_{i=1}^k i^3+ (k+1)^3\).
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