Find the sum of the following series :

0.6 + 0.66 + 0.666 + …. to n terms.

Let

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

Taking 6 as common we get

S = 6(0.1 + 0.11 + 0.111 + ...nterms)

Multiply and divide by 9

⇒ )

⇒ )

⇒

⇒

Now 1 + 1 + 1 + ..n = n

For 0.1 + 0.01 + 0.001 + ..nterms

∴ Common Ratio = r =

∴ Sum of GP for n terms = …(1)

⇒ a = 0.1, r = , n = n

∴ Substituting the above values in (1) we get

⇒

⇒

For second term the summation is n.

∴

⇒