SOLUTION: Can u please help me find the explicit formula for the series given the recursive formula ( a sub n= 3a sub (n-1)+1)

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Question 1111290: Can u please help me find the explicit formula for the series given the recursive formula ( a sub n= 3a sub (n-1)+1)
Found 2 solutions by math_helper, greenestamps:
Answer by math_helper(2461) About Me  (Show Source):
You can put this solution on YOUR website!

That depends, you have told us +a%5Bn%5D+=+3a%5Bn-1%5D+%2B+1+ but what is the value of +a%5B0%5D+? This is absolutely critical in finding the proper closed form solution.



Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


I was close to coming up with an explicit formula when you first posted this question; but things didn't quite seem to be working out for me.

Today when I went back and looked at the problem, things fell into place. It must have been my unconscious mind working on the problem for me....


Let the first term be a. Then the given recursive definition gives us

t%281%29+=+a
t%282%29+=+3a%2B1
t%283%29+=+9a%2B4
t%284%29+=+27a%2B13
t%285%29+=+81a%2B40
...

In the formula for t(n), the coefficient on a is clearly 3%5E%28n-1%29.

The constants in the formulas for the terms are

0 1 4 13 40 ...

A bit of experimentation, or perhaps some insight and logical analysis, shows the formula for this sequence to be %283%5E%28n-1%29-1%29%2F2.

So, given first term a, and with the given recursive definition, the formula for the n-th term of the sequence is

ANSWER: t%28n%29+=+%283%5E%28n-1%29%29a+%2B+%28%283%5E%28n-1%29%29-1%29%2F2