Find the sum of the terms of an infinite decreasing G.P. in which all the terms are positive, the first term is 4, and the difference between the third and fifth term is equal to 32/81.

Let a denote the first term of GP and r be the common ratio.


We know that nth term of a GP is given by-


an = arn-1


As, a = 4 (given)


And a5 – a3 = 32/81 (given)


4r4 – 4r2 = 32/81


4r2(r2 – 1) = 32/81


r2(r2 – 1) = 8/81


Let us denote r2 with y


81y(y-1) = 8


81y2 – 81y - 8 = 0


Using the formula of the quadratic equation to solve the equation, we have-


y =



y = 18/162 = 1/9 or y = 144/162 = 8/9


r2 = 1/9 or 8/9



As GP is decreasing and all the terms are positive so we will consider only those values of r which are positive and |r|<1


r =


Sum of infinite GP = ,where a is the first term and k is the common ratio.


Note: We can only use the above formula if |k|<1


the sum of respective GPs are –


S1 = {sum of GP for r = 1/3}


S2 = {sum of GP for r = (2√2)/3}


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