Question

Sum the series 5 + 55 + 555 + ... to n terms.

Answer 1

We have 5 + 55 + 555 + ... to n terms

= 5 $\times$ {1 + 11 + 111 + ... to n terms}

$=\frac{5}{9}$ $\times$ {9 + 99 + 999 + ... to n terms}

$=\frac{5}{9}\times \left\{\right(10-1)+({10}^2-1)+({10}^3-1)+\cdots \text{to n terms}\}$

$=\frac{5}{9}\times \left\{\right(10+{10}^2+{10}^3+\cdots \text{to n terms})-n\}$

$=\frac{5}{9}\times \{\frac{10\times ({10}^n-1)}{(10-1)}-n\}=\frac{5}{81}\times ({10}^{n+1}-9n-10)$

Hence, the required sum is $\frac{5}{81}\times ({10}^{n+1}-9n-10)$