+980 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.
+23252 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
