Chapter 5.3, Problem 46ES

In order for a proof by mathematical induction to be valid, the basis statement must be true for $n=a$ and the argument of the inductive step must be correct for every integer $K\ge a$ . In 45 and 46 find the mistakes in the "proofs" by mathematical induction.
46. “Theorem:” For every integer $n\ge 1,3^n-2$ is even.

“Proof (by mathematical induction): Suppose the theorem is true for an integer k, where $k\ge 1$ . That is, suppose that $3^k-2$ is even. We must show that $3^{k+1}-2$ is even. Observe that
$\begin{array}{l}{3}^{k+1}-2=3^k\cdot 3-2=3^k\left(1+2\right)-2\\ =\left(3^k-2\right)+3^k\cdot 2.\end{array}$

Now $3^k-2$ is even by inductive hypothesis and $3^k\cdot 2$ is even by inspection. Hence the sum of the two quantities is even (by Theorem 4.1.1). It follows that $3^{k+1}-2$ is even, which is what we needed to show."