The difference between the fourth term and the first term of a Geometric Progression is 52…

Question

The difference between the fourth term and the first term of a Geometric Progression is 52. If the sum of its first three terms is 26, then the sum of the first six terms of the progression is:

Accepted Answer

Let $a,ar,ar^2,\ldots$ be the G.P. $ar^3-a=52\Rightarrow a(r^3-1)=52$ ... (i)

$a(1+r+r^2)=26$ ... (ii)

Dividing (i) by (ii), using $r^3-1=(r-1)(1+r+r^2)$: $\dfrac{(r-1)(1+r+r^2)}{1+r+r^2}=r-1=2\Rightarrow r=3$, and from (ii): $a(1+3+9)=26\Rightarrow a=2$.

$$a(1+r+r^2+r^3+r^4+r^5)=a(1+r+r^2)(1+r^3)=26(1+27)=26\times28=728.$$

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