Question

Find the sum to n terms of the series 0.6+0.66+0.666+.....................

Answer 1

We have
$0.6+0.66+0.666+0.6666$

Let's take 6 out from the series
$6(0.1+0.11+0.111+0.1111+....$

Now multiply and divide by 9
$\frac{6}{9}(0.9+0.99+0.999+0.9999+....$

$\frac{6}{9}(1-0.1+1-0.01+1-0.001+1-0.0001+.... \\ \\\frac{6}{9}(1-10^{-1}+1-10^{-2}+1-10^{-3}+...$

all 1s add up to n
$\frac{6}{9}(n-\frac{1}{10}\times \frac{1-(\frac{1}{10})^{n+1}}{1-\frac{1}{10}})$