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Question

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

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Solution

(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 [ ( 101 )+( 10 2 1 )+( 10 3 1 )+.....ntimes ] = 5 9 [ ( 10+ 10 2 + 10 3 +....... )( 1+1+1+1...+ntimes ) ]

Further simplify.

S n = 5 9 [ 10( 10 n 1 ) 101 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 )+...nterms ] = 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 ).


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