21 lines
456 B
TeX
21 lines
456 B
TeX
\beginsection 1.3
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Prove $1^3+2^3+\cdots+n^3=(1+2+\cdots+n)^2$ for all natural numbers $n$.
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\medskip
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Induction Step 1: Show that $P_1$ is true.
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$$P_1=1^3=1^2$$
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Induction Step 2: Show that $P_n+(n+1)^3=P_{n+1}$.
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Note that $1+2+\cdots+n=n(n+1)/2$.
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$$\eqalign{
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P_n+(n+1)^3&=[n(n+1)/2]^2+(n+1)^3\cr
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&=(n+1)^2[(n/2)^2+(n+1)]\cr
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&=(n+1)^2(n^2/4+n+1)\cr
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\cr
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P_{n+1}&=[n(n+1)/2+(n+1)]^2\cr
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&=[(n+1)(n/2+1)]^2\cr
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&=(n+1)^2(n/2+1)^2\cr
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&=(n+1)^2(n^2/4+n+1)\cr
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}$$
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