I'm a bit confused regarding what the sequence is supposed to be. Is it correct to say that [imath]S_1 = 6[/imath], [imath]S_2 = 66[/imath] and so-on? And the problem is asking for a formula for the sum of all terms from [imath]S_1[/imath] through [imath]S_n[/imath] given some value for [imath]n[/imath]?
Assuming I interpret this correctly, my train of thought is as follows...
Each term in the sequence can be given with the following formula:
[imath]S_n = \frac{2}{3}(10^n-1)[/imath]
This isn't actually in the form of a
geometric series, as the terms are not separated by a common ratio. 6 to 66 is a ratio of 11, 66 to 666 is a ratio of [imath]10.\overline{09}[/imath], etc.
The running sum of terms is given with the following formula (given that [imath]\Sigma_0=0[/imath]):
[imath]\Sigma_n = \Sigma_{n-1} + S_n[/imath]
This is a
recurrence relation, and I never did learn how to work those out. Wolfram|Alpha
tells me that:
[imath]\Sigma_n = \frac{20}{27}(10^n-1) - \frac{2n}3[/imath]
This solves the problem as I understand it.
I'm interested to know how to resolve the formula properly--how to convert the recursive formula into something that is not recursive. I see this come up a lot in various contexts and it always flies over my head. In a pinch, I don't mind using Wolfram|Alpha for implementing solutions, but I like to know to do things "without a calculator" as it were. Any insights into this would be appreciated.