Divisibility Mathematical Induction Help

Question

Could you please help me with this question prove that $\displaystyle5^n + 2\cdot(11)^n$ is a multiple of $3$.

Thanks

Answer 1

Hints:

After you check for $\;n=1\;$ the inductive step is:

$$5^{n+1}+2\cdot 11^{n+1}=5\left(5^n+2\cdot 11^n\right)+6\cdot 2\cdot 11^n\ldots$$

Answer 2

Hint: $$\begin{align*}5^{k+1}+2\cdot11^{k+1}&=5\cdot5^k+2\cdot11\cdot11^k\\ &=5\cdot5^k+2\cdot(\color{red}5+\color{blue}6)\cdot11^k\\ &=5\cdot5^k+\color{red}{2\cdot5\cdot11^k}+\color{blue}{2\cdot6\cdot11^k}\\ &=5(5^k+\color{red}{2\cdot11^k})+\color{blue}{2\cdot6\cdot11^k}\\ &=5(5^k+\color{red}{2\cdot11^k})+\color{blue}{3\cdot2\cdot2\cdot11^k}\\\end{align*}$$

Answer 3

Base case: n = 0 $$5^0+2(11)^0 = 1 + 2 = 3$$ Inductive case: suppose the hypothesis holds for $n\le k$ $$5^{k+1} + 2(11)^{k+1}= 5 5^{k} + (6+5)2(11)^{k}=\cdots$$