28 lines
611 B
TeX
28 lines
611 B
TeX
\beginsection{1.4}
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\medskip
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(a) Guess a formula for $1+3+\cdots+(2n-1)$ by evaluating the sum for
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$n=1$, 2, 3, and 4. [For $n=1$, the sum is simply 1.]
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\medskip
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For each $n$ we have
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$$\eqalign{
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1&=1\cr
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1+3&=4\cr
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1+3+5&=9\cr
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1+3+5+7&=16\cr
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}$$
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therefore a reasonable guess would be
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$$1+3+\cdots+(2n-1)=n^2$$
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\medskip
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(b) Prove your formula using mathematical induction.
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\medskip
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We already have $n^2=1$ for $n=1$.
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For $n+1$ we have
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$$1+3+\cdots+(2n-1)+(2(n+1)-1)=n^2+2n+1=(n+1)^2$$
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Therefore the formula is true for $n+1$ whenever it is true for $n$.
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Hence by induction the formula is true for all $n\ge1$.
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