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I am not sure how to go about this "proof by induction" problem. below is my attempt.

Base Case: $n = 0$,substituting the value of $n$ to the equation $3^n+1$ $$= 3^0 + 1$$ $$= 1 + 1 = 2 $$ Thus the equation holds true for initial value of $n$ i.e $0$

Induction Hypothesis: Suppose the equation holds true for all the values of $n$: $1,2,3....k$ therefore, $3^k + 1$ results even.

Induction Step: $n = k$ holds true,
to prove: $3^{k+1}+ 1$ for $n = k+1$

LHS: $$3^{k+1}+ 1 = (3^k \cdot 3) + 1 = (3^k \cdot (2 + 1)) + 1 = 2 \cdot 3^k + 3^k + 1$$

The above solution results even, because since multiplying any integer with $2$ gives even integer, and from the Induction Hypothesis $3^k+1$ is even.

Hence $3^{k+1} + 1$ is even, thus $3^{n+1}$ is even for all values of $n\ge0$

Smit
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    Looks good. I think it feels odd because the problem is so artificial...I mean, you use the fact that Even + Even= Even yet you are supposed to pretend you don't notice that Odd + Odd = Even. But, that said, I think your argument is good. – lulu Feb 17 '17 at 13:47
  • @lulu you should make that an answer so this question can be closed. – Stella Biderman Feb 17 '17 at 14:31
  • @Smit What happens if $n=\sqrt2$? – Michael Rozenberg Feb 17 '17 at 14:53
  • @user236182 Exactly my point. It's a very artificial problem. – lulu Feb 17 '17 at 14:54
  • Honestly I think you should look for more interesting problems where induction doesn't seem artificial. This as others pointed out is indeed very trivial by noting that odd +1=even. – Xam Feb 17 '17 at 15:05
  • @lulu Thanks for the quick answer, I have changed the question to for all positive integers thus ur answer suits well now! @Michael Thanks for the pointer. – Smit Feb 17 '17 at 15:45
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    @lulu It's not artificial, because (at this level), proving $3^n$ odd requires induction on $n,,$ (e.g. using a product of odds is odd). – Bill Dubuque Feb 17 '17 at 18:27
  • Judging from the answers, you'd think that the OP just asked for a proof! Does no one care to critique/confirm the OP's work in an answer? – The Chaz 2.0 Feb 17 '17 at 22:09

3 Answers3

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Proof I

$$3^n + 1 \implies (2x + 1)+1 \implies 2x+2$$

Case: $n = 0$ $$ (2\cdot0 + 2) = 2 $$ Case: $n = n$ $$(2n + 2) = 2(n+1) $$ Case: $n = n+1$ $$(2(n+1) + 2) = 2n+4 = 2(n+2)$$ $$\Box$$

Proof II

$$(3^n + 1)\text{ mod }2 \implies (3^n + 2 -1) \text{ mod }2 \implies (3^n -1)\text{ mod }2 $$

Case: $n = 0$ $$3^0 \equiv 1 (\text{ mod }2)\implies 3^0 -1 \equiv 0 (\text{ mod }2)$$

Case: $n = n$ $$3^n \equiv 1 (\text{ mod }2)\implies 3^n -1 \equiv 0 (\text{ mod }2)$$ Case: $n = n+1$ $$3^{n+1} \equiv 1 (\text{ mod }2)\implies 3^{n+1} -1 \equiv 0 (\text{ mod }2)$$ $$\Box$$


Case of the Congruence Power Rule

Community
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Darío A. Gutiérrez
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Direct Proof:

$3^n$ has no factor of $2$, so it is odd. $3^n+1$ is one greater than an odd number.

Inductive Proof:

$3^0+1=2$ is even.

Suppose $3^n+1$ is even, then $$ 3^{n+1}+1=3\left(3^n+1\right)-2 $$ is an even number minus an even number, hence even.

robjohn
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An easier answer is to note that:

$3^n - 1 \equiv 1^n - 1 \equiv 0 (mod 2)$.Thus, we conclude that $2|(3^n -1)$. Hence, $3^n - 1$ is always even.