Question

Find the sum of the following series : 

0.6 + 0.66 + 0.666 + …. to n terms.

Answer 1

Let 

S = 0.6 + 0.66 + 0.666 + .....n terms 

Taking 6 as common we get 

S = 6(0.1 + 0.11 + 0.111 + ...n terms) 

Multiply and divide by 9

Now 1 + 1 + 1 + ..n = n 

For 0.1 + 0.01 + 0.001 + ..n terms 

∴ Common Ratio = r = \(\frac{0.01}{0.1} = \frac{1}{10}\)

∴ Sum of GP for n terms = \(\frac{a(r^n - 1)}{r-1}\) ... (1)

⇒ a = 0.1, r = \(\frac{1}{10}\), n = n

∴ Substituting the above values in (1) we get

For second term the summation is n.