recursive sequence formula

Question

Find an explicit formula for the recursive sequence $a(n+1)=3a(n)-8$ where $a(1)=4$.

Is the formula an=4 and how to prove it by induction I dont really know how to prove it using induction if this were the right answer

a(1)=4 .sorry i forgot to add it at the begining – Abdullah Alfaqir Jun 01 '17 at 21:52 — Abdullah Alfaqir, Jun 01 '17 at 21:52

score 6 · Answer 1 · answered Jun 01 '17 at 22:03

6

Hint (telescoping instead of induction): the recursion formula can be written as:

$$a_{n+1}-4=3(a_{n}-4)=3^2(a_{n-1}-4)=\cdots=3^{n}(a_1-4)$$

answered Jun 01 '17 at 22:03

dxiv

3

Hmmmm. That part that reads "$\cdots$". Isn't that shorthand for "induction"? – GEdgar Jun 01 '17 at 22:12
1

@GEdgar Telescoping is common enough that it has its own name, and is generally accepted without the formal base+step proof detail. Related: When do we use hidden induction? – dxiv Jun 01 '17 at 22:16
@GEdgar It's how I write my induction all the time :P (though I try to make it obvious) – Simply Beautiful Art Jun 01 '17 at 23:41

score 3 · Accepted Answer · answered Jun 01 '17 at 21:53

3

Let $P(n)$ be the statement $a(n) = 4$. Given $a(1) = 4$ and $P(1)$ is true.

Assume $a(k) = 4$ for some $k\in\mathbb N$. Then $$a(k+1) = 3a(k) - 8 = 12 - 8 = 4$$

So $P(k+1)$ is true.

answered Jun 01 '17 at 21:53

peterwhy

2 Answers2