Explicit formula for recurrence relation $a_{n+1} = 2a_n + 1$

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Solving a Recurrence Relation/Equation, is there more than 1 way to solve this?

How do I find an explicit formula for $a_n$ given $a_0 = 3$ and $a_{n+1} = 2a_n + 1$. I'm guessing it's probably related to the formula for first $n$ terms of a geometric series. Any help is appreciated

Duplicated: http://math.stackexchange.com/q/106036/23875 – Yai0Phah Jun 04 '12 at 02:57 — Yai0Phah, Jun 04 '12 at 02:57

score 3 · Answer 1 · answered Jun 04 '12 at 01:36

3

Write out the first few terms. Add one to each term. Make a conjecture. Prove it by induction.

answered Jun 04 '12 at 01:36

Gerry Myerson

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score 2 · Answer 2 · answered Jun 04 '12 at 01:39

2

Note that $a_{n+1}+1=2a_n+2=2(a_n+1)$. Define then $u_n=a_{n}+1$. Note that $u_{n+1}=2u_n$.

answered Jun 04 '12 at 01:39

Pedro

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19
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Explicit formula for recurrence relation $a_{n+1} = 2a_n + 1$

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