Prove the following by the principle of mathematical induction: 1… - Vidyadip

Question

Prove the following by the principle of mathematical induction: $11^{n+2}+12^{2 n+1}$ is divisible of $133$ for all $\text{n}\in\text{N}$

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Answer

Let $p(n): 11^{n+2}+12^{2 n+1}$ is divisible of 133 For $n=111^3+12^3=1331+1728=3059$ It is divisible of $133 \Rightarrow p(n)$ is true for $n=1$ Let $p(n)$ is true for $n=k$, so $11^{k+2}+12^{2 k+1}$ is divisible of $13311^{k+2}+12^{2 k+1}=133 \lambda \ldots$ (1)
We have to show that, $1^{k+3}+12^{2 k+3}$ is divisible of 133
Now, $11^{k+2} \cdot 11+12^{2 k+1} \cdot 12^2$ $=\left(133 \lambda-12^{2 \mathbf{k}+1}\right) 11+12^{2 \mathbf{k}+1} .144=11.133 \lambda+11.12^{2 \mathbf{k}+1}+144.12^{2 \mathbf{k}+1}=11.133 \lambda+133.12^{2 \mathbf{k}+1}$ $=133\left(11 \lambda+12^{2 \mathrm{k}+1}\right)=133 \mu \Rightarrow \mathrm{p}(\mathrm{n})$ is true for $\mathrm{n}=\mathrm{k}+1 \Rightarrow \mathrm{p}(\mathrm{n})$ is true for all by $\mathbf{n} \in \mathrm{N}$ PMI

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