Answer :

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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