Question

Let $S_n=\left(1\right)\left(5\right)+\left(2\right)\left(5^2\right)+\left(3\right)\left(5^3\right)+...+\left(n\right)\left(5^n\right)=\frac{1}{16}\left[\right(4n-1)5^a+b]$, then

• A. $a=n+1$
• B. $a=n$
• C. $b=5$
• D. $b=25$

Answer 1

Answer: (A), (C)

The correct options are
A $a=n+1$
C $b=5$

$S_n=\left(1\right)\left(5\right)+\left(2\right)\left(5^2\right)+\cdots +\left(n\right)5^n$.............. {i}
$5S_n=\left(1\right)(5{)}^2+(2\left)\right(5^3)+\cdots +(n-1)5^n+(n)5^{n+1}$............... {ii}
Subtracting {ii} from {i} by shifting one place, we obtain
$-4S_n=5+5^2+\cdots +5^n-n{5}^{n+1}=\frac{5(5^n-1)}{4}-n{5}^{n+1}$
$\therefore S_n=\frac{1}{16}\left[\right(n-1)5^{n+1}+5]$
On comparing we get $a=n+1$ and $b=5$