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The ratio of the sum of the first three terms is to that of the first 6 terms of a G.P. is 125 : 152. Find the common ratio.

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Given:

Sum of G.P of 3 terms is 125

By using the formula,

Sum of GP for n terms = a(rn – 1 )/(r – 1)

125 = a (rn – 1)/(r - 1)

125 = a (r3 – 1)/ (r - 1) … equation (1)

Now,

Sum of G.P of 6 terms is 152

By using the formula,

Sum of GP for n terms = a(rn – 1 )/(r – 1)

152 = a (rn – 1)/(r - 1)

152 = a (r6 – 1)/ (r - 1) … equation (2)

Let us divide equation (i) by (ii) we get,

125/152 = [a (r3 – 1)/ (r - 1)] / [a (r6 – 1)/ (r - 1)]

125/152 = (r3 – 1)/(r6 – 1)

125/152 = (r3 – 1)/[(r3 – 1) (r3 + 1)]

125/152 = 1/(r3 + 1)

125(r3 + 1) = 152

125r3 + 125 = 152

125r3 = 152 – 125

125r3 = 27

r3 = 27/125

r3 = 33/53

r = 3/5

∴ The common ratio is 3/5

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