# Can someone explain how $3^{n+1} + 3^n = 4 \cdot3^n$?

Slow morning. Can someone help me figure it out? I have a feeling it is trivially easy and not worthy of a thread. $$3^{n+1} + 3^n = 4\cdot3^n$$

Thanks.

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Did you mean $4\times 3^n$? I.e. $3^n$ instead of $3n$. –  Martin Sleziak Apr 22 '12 at 8:34
Yes. Edited. . . –  Mob Apr 22 '12 at 8:36
When I have a very slow morning, I use wolfram alpha ;-) –  dtldarek Apr 22 '12 at 8:43

## 2 Answers

Answers this version of the question

The previous version of the question claimed that for all $n \in \Bbb N$, $$3^{n+1}+3n=4\cdot 3n \tag{1}$$

However $(1)$ is not true for all $n$ as noted in the next part of the answer.

However what is true is: $$3^{n+1}+3^n=4\cdot3^n$$

To see this, note that $3^{n+1}=3^n \cdot 3$ and factor the $3^n$ out.

Answers the previous version of the question

Your claim is simply not true. For $n=2$, LHS equals $33$ while RHS evaluates to $24$.

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Christ. I am stupid. Thanks –  Mob Apr 22 '12 at 9:14
No you are not. But this feeling can make you one. So, please do not think this way, Regards, –  user21436 Apr 22 '12 at 9:26

Hint: Write $3^{n+1}$ as $3\cdot 3^n$, then factor $3^n$ out of the sum.

(I assume the question is about $3^{n+1}+3^n$.)

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