# Geometric series

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The sum of the first five terms of a geometric series is 186 and the sum of the first six terms is 378. if the fourth term is 48, determine a(first term),r(ratio), t10, S10.

• Geometric series -

if sum(6) = 378 and sum(5) = 186
then term(6) = 378-186 = 192

so
ar^5 = 192
ar^3 = 48
divide them
r^2 = 4
r = ±2
if r=2, then a(8) = 48 --->a = 6
if r = -2, then a(-8) = 48 --- a = -6

if a= 6, r=2, t(10) = 6(2^9) = 3072
if a= -6, r=-2 , t(10) = -6(-2)^9 = 3072

if a=6, r=2, sum(10) = 6(2^10 - 1)/1 = 6138
if a=-6,r=-2, sum(1) = -6((-2)^10 - 1)/-2-1) = 2046

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