SOLUTION: Find the general formula for the sum of this series: 3+33+333+3333+... to n terms

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Question 1087090: Find the general formula for the sum of this series:
3+33+333+3333+... to n terms
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!

Sn = 3+33+333+3333+... to n terms

If we multiply through by 3

3Sn = 9+99+999+9999+... to n terms

If we add 1 to each of the n terms on the right, we will 
have added n to both sides:

3Sn+n = 10+100+1000+10000+... to n terms

That is a geometric series with sum

 %28a%5B1%5D%28r%5En-1%29%29%2F%28r-1%29=%2810%2810%5En-1%29%29%2F%2810-1%29%22%22=%22%22expr%2810%2F9%29%2810%5En-1%29

So we have

3S%5Bn%5D%2Bn%22%22=%22%22expr%2810%2F9%29%2810%5En-1%29

Solve for Sn

3S%5Bn%5D%22%22=%22%22expr%2810%2F9%29%2810%5En-1%29-n

S%5Bn%5D%22%22=%22%22expr%2810%2F27%29%2810%5En-1%29-n%2F3

Edwin