Question

If $5+55+555+\cdots \text{to}n\text{terms}=\frac{5}{81}({10}^{n+1}+x+yn)$, then the value of $y-x$ is

• A. $1$
• B. $-19$
• C. $-1$
• D. $19$

Answer 1

The correct option is A $1$

Let $S_n$ denote the sum to $n$ terms of the given series, then
$S_n=5+55+555+\cdots \text{to}n\text{terms}\Rightarrow S_n=5[1+11+111+\cdots \text{to}n\text{terms}]\Rightarrow S_n=\frac{5}{9}[9+99+999+\cdots \text{to}n\text{terms}]\Rightarrow S_n=\frac{5}{9}\left[\right(10-1)+({10}^2-1)+({10}^3-1)+\cdots ({10}^n-1\left)\right]\Rightarrow S_n=\frac{5}{9}\left[\right(10+{10}^2+{10}^3+\cdots {10}^n)-n]\Rightarrow S_n=\frac{5}{9}[10\times \frac{({10}^n-1)}{10-1}-n]\Rightarrow S_n=\frac{5}{9}[\frac{10}{9}\times ({10}^n-1)-n]\Rightarrow S_n=\frac{5}{81}[{10}^{n+1}-10-9n]\Rightarrow x=-10,y=-9\therefore y-x=1$