SOLUTION: Calculate the sum of the first 6 terms of the geometric series with the 3rd term 6 and the common ratio 2/3. A) 1330/243 B) 211/6 C) 665/18 D) 665/729

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Question 1173922: Calculate the sum of the first 6 terms of the geometric series with the 3rd term 6 and the common ratio 2/3.
A) 1330/243
B) 211/6
C) 665/18
D) 665/729
Found 2 solutions by greenestamps, MathTherapy:
Answer by greenestamps(13200) (Show Source):
You can put this solution on YOUR website!

For the sum of a geometric series with 6 terms and "nice" numbers, finding the numbers and adding them is easier than using the formula for the sum of a finite geometric series.

3rd term 6 and common ratio 2/3 means...
3rd term 6
2nd term 6*(3/2) = 9
1st term 9*(3/2) = 27/2 = 13 1/2
4th term 6*(2/3) = 4
5th term 4*(2/3) = 8/3 = 2 2/3
6th term (8/3(*(2/3) = 16/9 = 1 7/9

With the given answer choices, we can almost choose the right answer without adding the terms. The denominators of answer choices A and D are powers of 3; those answers are not possible because the 1st term is 27/2.

So the answer is either B or C. Converting those answer choices to mixed numbers gives
B: 35 1/6
C: 36 7/18

Adding the 6 terms....
13 1/2 + 9 = 22 1/2
22 1/2 + 6 = 28 1/2
28 1/2 + 4 = 32 1/2
32 1/2 + 2 2/3 = 35 1/6

That is already answer B; but it is the sum of just the first 5 terms. So the answer has to be choice C. Checking that answer choice....

35 1/6 + 1 7/9 = 35 3/18+1 14/18 = 36 17/18

Yes, answer choice C is the correct sum.

Answer by MathTherapy(10552) (Show Source):
You can put this solution on YOUR website!

Calculate the sum of the first 6 terms of the geometric series with the 3rd term 6 and the common ratio 2/3.
A) 1330/243
B) 211/6
C) 665/18
D) 665/729

Formula for a GP term:
Substituting 3 for n, we get:
Substituting we get:
Formula for sum of a GP:
Substituting 6 for n, we get:
Substituting we get: