SOLUTION: Find S(with a sideways 8 beside it) if an=27(1/3)^n, n=1,2,3,4,... I have no idea how to do this problem. Please help me.

Algebra ->  Sequences-and-series -> SOLUTION: Find S(with a sideways 8 beside it) if an=27(1/3)^n, n=1,2,3,4,... I have no idea how to do this problem. Please help me.      Log On


   

Question 37827: Find S(with a sideways 8 beside it) if an=27(1/3)^n, n=1,2,3,4,... I have no idea how to do this problem. Please help me.
Answer by longjonsilver(2297) About Me  (Show Source):
You can put this solution on YOUR website!
OK...

+a%5Bn%5D+=+27%2A%281%2F3%29%5En+ , n=1,2,3,4,...

so putting a few values of n into this formula, we get:

+a%5B1%5D+=+27%2A%281%2F3%29%5E1+
+a%5B1%5D+=+27%2A%281%2F3%29+
+a%5B1%5D+=+9+

+a%5B2%5D+=+27%2A%281%2F3%29%5E2+
+a%5B2%5D+=+27%2A%281%2F9%29+
+a%5B2%5D+=+3+

+a%5B3%5D+=+27%2A%281%2F3%29%5E3+
+a%5B3%5D+=+27%2A%281%2F27%29+
+a%5B3%5D+=+1+
etc

So we have a GP: 9, 3, 1, ....
whose common ratio is 1/3.

Now formula for "sum to infinity" is:

+S+=+a%2F%281-r%29+
so, +S+=+9%2F%281-%281%2F3%29%29+
+S+=+9%2F%282%2F3%29+
+S+=+27%2F2+
S = 13.5

which you can begin to see if you add a few of the terms up yourself:
9+3+1+(1/3)+...
is already 13.333333

jon.