Deriving a formula and induction

AI Thread Summary
The discussion centers on deriving a formula for the recursive sequence defined by T(2)=1 and T(n+1)=1+2T(n). Participants express confusion about how to derive a new formula from the given recursive definition. One user identifies a pattern in the sequence, suggesting that T(n) can be expressed as 2^n - 1. The conversation highlights the need for clarity in transitioning from the recursive definition to a closed-form expression. Ultimately, the goal is to derive the formula and prove it through mathematical induction.
muso07
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Homework Statement


We know that T(2)=1, and T(n+1)=1+2T(n), i.e. T(3)=1+2*1=3, etc.
Derive a formula for T(n) from above information. Prove the formula by mathematical induction.

Homework Equations


??
I don't think there are any apart from the stuff from above.

The Attempt at a Solution


I don't really know where to start... Isn't T(n+1)=1+2T(n) already a formula? How am I supposed to come up with another one?

Basically, in the previous part, I showed that the sequence goes 1, 3, 7, 15, 31,.. as per the rule. (Not sure what the point of that was.) But the question says to use that to derive the formula.

I just need some help with coming up with the formula.. hopefully I can do the induction stuff by myself.

Any help would be greatly appreciated!
 
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Hi Muso07

I noticed you could re-write the first 2 terms as:
T(2) = 1 = 1+(1-1) = 2-1= 2^1 - 1
T(3) = 1+2.1 = 2 + 2 - 1 = 2^2 - 1
hope this helps
 
Thank you so much!
 

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