Question

Let $x, \, y$ and $z$ be the respective sum of the first $n$ terms, the next $n$ terms and the next $n$ terms of a geometric progression, then $x, \, y, \, z$ are in

• A. arithmetic progression
• B. geometric progression
• C. harmonic progression
• D. None of these

Answer 1

Answer: (B)

Let $A$ be the 1^st term and $R$ be the common ratio of the given G.P.
Sum of the first $n$ terms of the G.P. is $x \, =\frac{A \left(\right. 1 - R^{n} \left.\right)}{1 - R}$ . The next $n$ terms form a G.P. series of $n$ terms with the first term as the $\left(n + 1\right)^{t h}$ term of the given G.P. is given by $t_{n + 1}=AR^{n}$ . Sum of the next $n$ terms of the G.P. is $y \, ⇒ \, \, y \, =\frac{A R^{n} \left(\right. 1 - R^{n} \left.\right)}{1 - R}$ and
$z=AR^{2 n}\frac{\left(1 - R^{n}\right)}{1 - R}$ $⇒ \, \, y^{2}=zx$
$∴ \, \, $ $x$ , $y$ , $z$ are in G.P.