The sum of an infinitely decreasing GP is 3.5 and the sum of squares in terms is 14716. The sum of the cubes of the terms of the progression is _____ (find the nearest integer)

a+ar+ar2+ar3+........=3.5 and a2+a2r2+a2r4+a3r6+...........=14716∴a1−r=72 and a21−r2=14716Solve above two equations a = 3, r = 17a3+a3r3+a3r6+......=a31−r3=102938=(27.07)

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The sum of an infinitely decreasing GP is 3.5 and the sum of squares in terms is $\frac{147}{16}$ .
The sum of the cubes of the terms of the progression is _____ (find the nearest integer)

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answer is 27.

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Detailed Solution

$a + a r + a r^{2} + a r^{3} + ........ = 3.5$ and $a^{2} + a^{2} r^{2} + a^{2} r^{4} + a^{3} r^{6} + ........... = \frac{147}{16}$

$∴ \frac{a}{1 - r} = \frac{7}{2}$ and $\frac{a^{2}}{1 - r^{2}} = \frac{147}{16}$

Solve above two equations a = 3, r = $\frac{1}{7}$

$a^{3} + a^{3} r^{3} + a^{3} r^{6} + ...... = \frac{a^{3}}{1 - r^{3}} = \frac{1029}{38} = (27.07)$

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