Answer: Option B
:
B
Conventional Method
From the data given in the question
$S_3=16=\frac{a(r^3-1)}{r-1}$................................(1)
$S_6=16+128=144=\frac{a(r^6-1)}{r-1}=\frac{a(r^3-1)(r^3+1)}{r-1}$................................(2)
9 = (r${}^3$ + 1)
Hence r =2
Substituting in equation 1, we get
a = $\frac{16}{7}$
We know that,
$S_n=\frac{a(r^n-1)}{r-1}$
Substituting the values of a and r found above, we get
$S_n=\frac{16}{7}(2^n-1)$
Thus, answer is option (b)
Shortcut-
Substitution The answerto this question is in the question itself!
At n=3, the sum should be 16
Hence, in the correct answer option, at n=3, we should get 16
Option (a)= $\frac{16}{7}$(2${}^3$+ 1)$\ne$16
Option(b)=$\frac{16}{7}$(2${}^3$-1) = 16
Option (c)=$\frac{9}{7}$(3${}^3$-1)$\ne$16
Option (d)=$\frac{16}{7}$(3${}^3$ -1)$\ne$16
Options (a), (c) and (d) can never be the answer. Answer is option (b)