Mathematical Induction: Divisibility by 7 Example

October 25, 2014 | 1 min read

Prove that \(3^{2n+1}+2^{n+2}\) is divisible by \(7\) for every nonnegative integer \(n\).

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Let \(P(n)\) be: \(7\mid3^{2n+1}+2^{n+2}\).

We wish to show that \(P(n)\) is true \(\forall n \geq0\).

We first verify the base \(n=0\) case: \[3^{2\times0+1}+2^{0+2}=3^1+2^2=7\] \[1 \times 7 = 7\]

We wish to show that \(P(k)\implies P(k+1) \, \forall k \geq 0\).

We assume \(P(k)\) to be true for some \(k \geq 0\).

Thus \(7q=3^{2k+1}+2^{k+2}\) for some \(q \in \mathbb{Z}\).

Consider

\[\begin{align*} & 3^{2(k+1)+1}+2^{(k+1)+2} \\ &= 3^{(2k+1)+2}+2^{(k+2)+1} \\ &= 9(3^{2k+1})+2(2^{k+2}) \\ &= 9{(3^{2k+1}+2^{k+2})}-7(2^{k+2}) \end{align*}\]

We have shown \(3^{2(k+1)+1}+2^{(k+1)+2}=9(3^{2k+1}+2^{k+2})-7(2^{k+2})\).

Because \(7\mid3^{2k+1}+2^{k+2}\) as assumed, and \(7\mid7\) trivially, it follows that \(7\mid3^{2(k+1)+1}+2^{(k+1)+2} \).

Thus, \(P(k)\implies P(k+1) \) as needed, so \(P(n) \) is true \(\forall n \geq 0\)