SOLUTION: In a geometric series, the sum of the first three terms is 304, and the sum of the first six terms is 1330. Find the sum of the first seven terms.

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Question 898542: In a geometric series, the sum of the first three terms is 304, and the sum of the first six terms is 1330. Find the sum of the first seven terms.
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
In a geometric series, the sum of the first three terms is 304,
a%5B1%5D%2Ba%5B1%5Dr%2Ba%5B1%5Dr%5E2%22%22=%22%22304

(1)   a%5B1%5D%281%2Br%2Br%5E2%29%22%22=%22%22304

and the sum of the first six terms is 1330.
So the sum of the 3rd, 4th and 5th terms only is 1330-304 = 1026

a%5B1%5Dr%5E3%2Ba%5B1%5Dr%5E4%2Ba%5B1%5Dr%5E5%22%22=%22%221026

(2)   a%5B1%5Dr%5E3%281%2Br%2Br%5E2%29%22%22=%22%221026

Dividing equals by equals, equation (2) by equation (1)

    %28a%5B1%5Dr%5E3%281%2Br%2Br%5E2%29%29%2F%28a%5B1%5D%281%2Br%2Br%5E2%29%29%22%22=%22%221026%2F304

    %22%22=%22%2227%2F8

    r%5E3%22%22=%22%2227%2F8

Taking cube roots of both sides

    r%22%22=%22%223%2F2

Substitute in equation (1)

(1)   a%5B1%5D%281%2Br%2Br%5E2%29%22%22=%22%22304

      a%5B1%5D%281%2B3%2F2%2B%283%2F2%29%5E2%29%22%22=%22%22304

      a%5B1%5D%281%2B3%2F2%2B9%2F4%29%22%22=%22%22304

Multiply the terms in the parentheses and the right side by 4
to clear of fractions:

      a%5B1%5D%284%2B6%2B9%29%22%22=%22%221216

      a%5B1%5D%2819%29%22%22=%22%221216

Divide both sides by 19

      a%5B1%5D%22%22=%22%221216%2F19

      a%5B1%5D%22%22=%22%2264

Find the sum of the first seven terms.
S%5Bn%5D%22%22=%22%22%28a%5B1%5D%28r%5En-1%29%29%2F%28r-1%29

S%5B7%5D%22%22=%22%22%2864%28%283%2F2%29%5E7-1%29%29%2F%283%2F2-1%29

S%5B7%5D%22%22=%22%22%2864%282187%2F128-1%29%29%2F%283%2F2-1%29

Multiply the top out, the 16 goes into the 128 2 times,
and write the 1 on the bottom as 2%2F2

S%5B7%5D%22%22=%22%22%282187%2F2-64%29%2F%283%2F2-2%2F2%29

S%5B7%5D%22%22=%22%22%282187%2F2-64%29%2F%281%2F2%29

Multiply top and bottom by 2

S%5B7%5D%22%22=%22%222%282187%2F2-64%29%2F%282%2Aexpr%281%2F2%29%29
 
S%5B7%5D%22%22=%22%22%282187-128%29%2F1

S%5B7%5D%22%22=%22%222059

Edwin