Answer:
Sn = (7 -(6n +7)3^-n)/4
Explanation:
You want the sum of n terms of the series 2/3 +5/9 +8/27 +11/81 +....
Partial sums
The sums of n terms for n = 1 .. 5 are ...
2/3, 11/9, 41/27, 134/81, 416/243
We recognize the denominators as powers of 3. The numerator sequence is ...
2, 11, 41, 134, 416, ...
Differences are ...
9, 30, 93, 282
And second differences are ...
21, 63, 189
We note that these have a common ratio of 3, leading us to speculate that the numerator sequence can be written as the sum of a linear expression and a power of 3.
Coefficients
Assuming the sequence of numerator terms is of the form ...
a·3^n +bn +c
We can write equations for a, b, c:
a·3 +b·1 +c = 2 . . . . . . for n = 1
a·9 +b·2 +c = 11 . . . . . .for n = 2
a·27 +b·3 +c = 41 . . . . for n = 3
These equations can be solved for a, b, c using any of your favorite techniques. The attached calculator output shows the values to be ...
a = 1.75 = 7/4
b = -1.5 = -6/4
c = -1.75 = -7/4
Nth sum
Then the sum of n terms can be written as ...
__
Additional comment
The n-th term of the series is (3n-1)/3^n. This can be decomposed into two sums: 3(n/3^n) -1/3^n. The latter is the well-known sum of a geometric series. The sum of the first term is less well-known: ∑(k/3^k) = (3(3^n-1)-2n)/(4·3^n).