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An infinite geometric series has a first term of 12 and a second term of 4. A second infinite geometric series has the same first term of 12, a second term of 4+n, and a sum of four times that of the first series. Find the value of n.

 Apr 19, 2020
 #1
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Sum of an infinite series:  Sum  =  a / (1 - r)

 

First series:  common ratio =  4 / 12  =  1/3      --->     Sum  =  12 / (1 - 1/3)  =  12 / (2/3  =  18

 

Second series:  common ratio  =  (4 + n) / 12     --->     Sum  =  12 / ( 1 - (4 + n)/12 )

--->     12 / ( 1 - (4 + n)/12 )  =  4 · 18

--->     12 / ( 1 - (4 + n)/12 )  =  72

multiplying the numerator and denominator on the left side by 12/12

--->     144 / ( 12 - (4 + n) )  =  72

--->     144 / ( 8 - n )  =  72

--->     144  =  72(8 - n)

--->     144  =  576 - 8n

--->    -432  =  -8n

--->         n  =  6

 Apr 20, 2020

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