Question

In a sequence if the sum of the first $n$ terms is given by $S_n=2^{np}-1$ where ' $p$ ' is fixed real number. The nature of the sequence, is

• A. A.P
• B. G.P
• C. H.P
• D. A.G.P

Answer 1

Answer: (B)

$t_{n+1}=S_{n+1}-S_n=\left(2^{(n+1)p}-1\right)-\left(2^{np}-1\right)=2^{np}\cdot 2^p-2^{np}=2^{np}\left(2^p-1\right)$
$|||ly{t}_n=S_n-S_{n-1}=\left(2^{np}-1\right)-\left(2^{(n-1)p}-1\right)=2^{(n-1)p}\left(2^p-1\right)$
$\therefore \frac{t_{n+1}}{t_n}=\frac{2^{np}}{2^{(n-1)p}}=2^p=\text{ constant }$
Hence sequence is G.P.