your first series has
a = 12, r = 4/12 = 1/3
and the sum of all the terms is 12/(1-1/3) = 18
your second has a = 12, and r = (4+n)/12
the sum of all the terms = 12/(1 - (4+n)/12)
= 12/( (12 - 4 - n)/12)
= 144/(8 - n)
but you told us that this is 4(18)
144/(8-n) = 72
144 = 576 - 72n
72n = 432
n = 6
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 $n.$
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