Question

The sum of the series 6 + 66 + 666 + ........ upto n terms is

• A. (10^{n - 1} - 9n + 10)/ 81
• B. 2(10^{n + 1} - 9n - 10)/ 27
• C. 2(10ⁿ - 9n - 10)/ 27
• D. None of these

Answer 1

The correct option is B 2(10^{n + 1} - 9n - 10)/ 27

Given series $6+66+666+.....+$ upto n terms

$=\frac{6}{9}(9+99+999+....$ upto n terms)

$=\frac{2}{3}(10+{10}^2+{10}^3+.....+$ upto n terms -n)

$=\frac{2}{3}(\frac{10({10}^n-1)}{10-1}-n)=\frac{1}{27}\left[20\right({10}^n-1)-{18}_n]$

$=\frac{2\left({10}^{n+1}\right)-9_n-10}{27}$

Trick : Put $n=1$