Use mathematical induction to prove that \(3^n+7^n\) is divisible by 10 for all odd \(n\). (3 marks)
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Use mathematical induction to prove that \(3^n+7^n\) is divisible by 10 for all odd \(n\). (3 marks)
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\(\text{Proof (See Worked Solutions)}\)
\(\text{Prove \(3^n+7^n\) is divisible by 10 for \(n\) odd.}\)
\(\text{Prove true for}\ \ n=1:\)
\(3^1+7^1=10 \ \ \text {(which is divisible by 10)}\)
\(\therefore \ \text{True for} \ \ n=1\)
\(\text{Assume true for} \ \ n=k\) :
\(3^k+7^k=10 P \ \text{(where \(P\) is an integer)}\)
\(\Rightarrow 3^k=10 P-7^k\ \ldots\ (1)\)
\(\text{Prove true for}\ \ n=k+2:\)
| \(3^{k+2}+7^{k+2}\) | \(=9 \times 3^k+49 \times 7^k\) |
| \(=9\left(10 P-7^k\right)+49 \times 7^k\) | |
| \(=90 P-9 \times 7^k+49 \times 7^k\) | |
| \(=90P+40 \times 7^k\) | |
| \(=10\left(9 P + 4 \times 7^k\right)\) |
\(\Rightarrow \ \text{True for} \ \ n=k+2\)
\(\therefore \ \text{Since true for \(n=1\), by PMI, true for integers \(n\) odd}.\)