Induction Watch

Planto
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#1
Report Thread starter 9 years ago
#1
Show that 1^3 + 2^3 + 3^3 + ... + k^3 = (1 + 2 + 3 + ... + k)^2

Base step:

Let k = 1:
1^3 = 1^2

Inductive hypothesis: k^3 + (k + 1)^3 = (2k + 1)^2

What's the most elegant way of showing this?
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Oh I Really Don't Care
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#2
Report 9 years ago
#2
hhmmm. The above does not look right. You wil want to quote [or easily prove] the summation of the natural numbers 1 to k and form the inductive hypotheses;

 1^3 + 2^3 +... + k^3 = \left(\frac{1}{2}k(k+1)\right)^2

add (k + 1)^3 to both sides - try factoring the rhs to show that P(k) implies truth of p(k+1)

You have established that P(1) is true.
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Planto
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#3
Report Thread starter 9 years ago
#3
Ah yes, it never occurred to me to look at it that way. Thanks. I see the epic fail I made with forming my induction hypothesis now :o:

Although, I've got this hunch that all we're going to have to prove in the exam is \displaystyle\sum_{i=1}^{n} x_{i} = \dfrac{n(n+1)}{2} which should be straightforward enough.
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Unbounded
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#4
Report 9 years ago
#4
(Original post by Planto)
Although, I've got this hunch that all we're going to have to prove in the exam is \displaystyle\sum_{i=1}^{n} x_{i} = \dfrac{n(n+1)}{2} which should be straightforward enough.
If this is FP1, I would have thought you could assume this, and it's in the formula booklet, isn't it?
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Planto
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#5
Report Thread starter 9 years ago
#5
Nah, this is my level 1 Computer Science maths exam. It's a whole lot of content squeezed into one module, so most of the questions are just to confirm comprehension of principle, rather than any major problem solving tasks.
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