Question

Find the sum of the following series up to n terms: (i) 5 + 55 + 555 + … (ii) .6 +.66 +. 666 +…

Answer 1

(i)

The given series is 5+55+555+5555+.........

Let, S n =5+55+555+5555+.....

Divide and Multiply S n by 9,

S n =5+55+555+5555+... = 5 9 ( 9+99+999+9999+.... ) = 5 9 [ ( 10−1 )+( 10 2 −1 )+( 10 3 −1 )+.....n times ] = 5 9 [ ( 10+ 10 2 + 10 3 +....... )−( 1+1+1+1...+n times ) ]

Further simplify.

S n = 5 9 [ 10( 10 n −1 ) 10−1 −n ] = 5 9 [ 10( 10 n −1 ) 9 −n ] = 50 81 ( 10 n −1 )− 5n 9

Thus, the sum of the given series is 50 81 ( 10 n −1 )− 5n 9 .

(ii)

The given series is .6+.66+.666+.6666+....

Let, S n =.6+.66+.666+.6666+.....

Divide and Multiply S n by 9,

S n =.6+.66+.666+.6666+.... = 6 9 ( 0.9+0.99+0.999+0.9999+.... ) = 6 9 [ ( 1− 1 10 )+( 1− 1 ( 10 ) 2 )+( 1− 1 ( 10 ) 3 )+...n terms ] = 6 9 [ ( 1+1+1+1...+n )− 1 10 ( 1+ 1 10 + 1 ( 10 ) 2 +..... ) ]

Further simplify.

S n = 2 3 [ n− 1 10 ( 1− ( 1 10 ) n 1− 1 10 ) ] = 2 3 n− 2 30 × 10 9 ( 1− 10 −n ) = 2 3 n− 2 27 ( 1− 10 −n )

Thus, the sum of the given series is 2 3 n− 2 27 ( 1− 10 −n ).