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Find the sum to n terms (i) 1 + (1)/(...

Find the sum to n terms
`(i) 1 + (1)/(3) + (1)/(9)+...`
(ii) 5 + 55 + 555+ ....

Text Solution

Verified by Experts

(i) Here , ` a = 1 ,r = (1)/(3)`
`therefore S_(n) = (a(1-r^(n)))/(1-r) = (1xx[1-((1)/(3))^(n)])/(1-(1)/(3)) = (3)/(2)(1 - (1)/(3^(n)))`
(ii) Let `S_(n) = 5 + 55 + 555 + …` to n terms
= ` 5 (1 + 1 1 + 111 + ` … to n terms ) ` (5)/(9) [9 + 99 + 999 + ....` to n terms]
` (5)/(9) [ 10 + 100 + 1000 + ...` to n terms ) = ` (1 + 1 + ...` to n terms )]
`(5)/(9) [ 10 ((10^(A) - 1))/( 10 - 1) - n] (5)/(81) (10^(n+1) - 9n - 10)`
` therefore S_(n) = (5)/(81)(10^(n+1) - 9n - 10)` .
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