Find the sum of the following series :

0.5 + 0.55 + 0.555 + …. to n terms

Let

S = 0.5 + 0.55 + 0.555 + .....n terms

Taking 5 as common we get,

S = 5(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.

∴

⇒