The sum of an infinite geometric series is 36. The second term of the series is 8. Find two possibilities for combinations of a (the first term) and r (the common ratio).

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- February 21st 2007, 06:10 PMMr_GreenGeometric series and such..
The sum of an infinite geometric series is 36. The second term of the series is 8. Find two possibilities for combinations of a (the first term) and r (the common ratio).

Thanks - February 21st 2007, 06:21 PMAryth
My answer is wrong... I realized that... The 2nd term on both of those was not 8. Sorry.

- February 21st 2007, 06:49 PMSoroban
Hello, Mr_Green!

Quote:

The sum of an infinite geometric series is 36.

The second term of the series is 8.

Find two possibilities for combinations of a (the first term) and r (the common ratio).

Given the first term*a*and the common ratio*r*,

. . the nth term is: .a_n .= .ar^{n-1}

. . and the sum is: .S .= .a/(1 - r)

We are told the second term is 8: .a_2 .= .ar .= .8 . → . r = 8/a .**[1]**

We are told the sum is 36: .S .= .a/(1 - r) = 36 . → . a .= .36 - 36r .**[2]**

Substitute [1] into [2]: .a .= .36 - 36(8/a)

We have a quadratic: .aČ - 36a + 288 .= .0

. . which factors: .(a - 12)(a - 24) .= .0

. . and has roots: .a .= .12, 24

Substitute into [1] and we get: .r .= .2/3, 1/3

Therefore, the series are: .a = 12, r = 2/3 .and .a = 24, r = 1/3