Date: Tue, 24 Aug 1999 03:47:48 -0600 (CST)
Subject: How to prove it?
Name: Bernard
Who is asking: Teacher
Level: Secondary
Question:
How to prove 13 + 23 + 33 + 43 + ... n3 is equal to (1+2+3+...n)2? (for n is positive integer),
thank you.
Hi Bernard
There is a proof using mathematical induction. It uses the fact that for any positive integer n,
1+2+3+...+n = n(n+1)/2
Proof by mathematical induction.
When n = 1 the result is clear, 13 = 12
Assume the result is true for n = k, that is
13 + 23 + 33 + 43 + ... k3 = (1 + 2 + 3 +...k)2
Let n = k + 1, then
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13 + 23 + 33 + 43 + ... n3 |
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= 13 + 23 + 33 + 43 + ... + (k+1)3 |
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= (1 + 2 + 3 + ... + k)2 + (k+1)3 |
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= (k(k+1)/2)2 + (k+1)3 |
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= (k2 (k+1)2)/4 + (k+1)3 |
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= (k+1)2/4 (k2 + 4k + 4) |
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= ((k+1)2 (k+2)2)/4 |
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= ((k+1)(k+2)/2)2 |
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= (1 + 2 + 3 + ... + (k+1))2 |
Thus 13 + 23 + 33 + 43 + ... n3 = (1+2+3+...n)2 for all positive intergers n. Cheers
Harley
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