Question 1121118: Let {an}∞n=1 be a sequence whose partial sums are {Sn}∞n=1. Suppose that a1=2 and an=4⋅an−1.
Find a general formula for the nth term of the sequence of partial sums.
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!
I will ignore the first sentence of your post, since the notation is not anything I have seen....
The question you ask can still be answered.
The sequence has first term 2, and each subsequent term is 4 times the preceding term. So the sequence is
2, 8, 32, 128, 512, ...
The partial sums are
2, 10, 42, 170, 682, ...
The first partial sum can be written as
2 = 2(1) = 2(4^0)
The second partial sum can be written as
10 = 2(1+4) = 2(4^0+4^1)
The third partial sum can be written as
42 = 2(1+4+16) = 2(4^0+4^1+4^2)
The fourth partial sum can be written as
170 = 2(1+4+16+64) = 2(4^0+4^1+4^2+4^3)
The n-th partial sum can be written as
2(4^0+4^1+4^2+...+4^(n-1))
The expression in parentheses is a geometric series, for which there is a nice closed form for the sum.
The general formula for the n-th partial sum of the given sequence is
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