Find the sum of the following series :

9 + 99 + 999 + … to n terms.

Taking 9 in common we get

9(1 + 11 + 111 + ....n)

Now Multiply and Divide by 9 we get,

⇒ )

⇒ )

⇒

⇒

Now first term is in GP.

10, 100, 1000…to n terms

∴ Common Ratio = r =

∴ Sum of GP for n terms = …(1)

⇒ a = 10, r = 10, n = n

∴ Substituting the above values in (1) we get

⇒

⇒

For the second term the summation is n.

∴

⇒ .