Question

Use an induction proof to prove this statement: For n≥1, 4^n+5 is divisible by 3.

Answer 1

Answer:

See below

Step-by-step explanation:

We shall prove that for all $n\in\mathbb{N},3|(4^n+5)$. This tells us that 3 divides 4^n+5 with a remainder of zero.

If we let $n=1$, then we have $4^{1}+5=9$, and evidently, $9|3$.

Assume that $4^n+5$ is divisible by $3$ for $n=k, k\in\mathbb{N}$. Then, by this assumption, $3|(4^n+5)\Rightarrow4^k+5=3m,\: m\in\mathbb{Z}$.

Now, let $n=k+1$. Then:

$4^{k+1}+5=4^k\cdot4+5\\=4^k(3+1)+5\\=3\cdot4^k+4^k+5\\=3\cdot4^k+3m\\=3(4^k+m)$

Since $3|(4^k+m)$, we may conclude, by the axiom of induction, that the property holds for all $n\in\mathbb{N}$.