19 lines
360 B
TeX
19 lines
360 B
TeX
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\beginsection 1.2
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Prove $3+11+\cdots+(8n-5)=4n^2-n$ 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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$$P_1=(8\cdot1-5)=3$$
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Induction Step 2: Show that $P_n+(8(n+1)-5)=P_{n+1}$.
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$$\eqalign{
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P_n+(8(n+1)-5)&=4n^2-n+8n+3\cr
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&=4n^2+7n+3\cr
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\cr
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P_{n+1}&=4(n+1)^2-(n+1)\cr
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&=4(n^2+2n+1)-n-1\cr
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&=4n^2+8n+4-n-1\cr
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&=4n^2+7n+3\cr
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}$$
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