Latest Gate Aptitude Question SOLUTION: the sum of n terms of series 4+44+444+4444+44444+ ......is (1) (4/81)[10^(n+1)-9n-1] (2) (4/81)[10^(n-1)-9n-1] (3) (4/81)[10^(n+1)-9n-10] (4) (4/81)[10^n-9n-10]

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(#M40013857) GATE Numerical Ability question sum of series Keep an EYE

the sum of n terms of series 4+44+444+4444+44444+ ......is
(1) (4/81)[10^(n+1)-9n-1]
(2) (4/81)[10^(n-1)-9n-1]
(3) (4/81)[10^(n+1)-9n-10]
(4) (4/81)[10^n-9n-10]

Asked In Gate sc mandota (11 years ago)

Solved Dr.Dipin Singh Read Solution (5)

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Read Solution (5) :

Dr.Dipin Singh  (11 years ago)

(3) (4/81)[10^(n+1)-9n-10]

4 + 44 + 444 + ... to n terms

= 4 [ 1 + 11 + 111 + ... to n terms ]

= (4/9) [ 9 + 99 + 999 + ... to n terms ]

= (4/9) [ (10 - 1) + (10^2 - 1) + (10^3 - 1) + ... + (10^n - 1) ]

= (4/9) [ ( 10 + 10^2 + 10^3 + ... + 10^n ) - ( 1 + 1 + 1 + ... n times ) ]

= (4/9) { 10 [ ( 10^n - 1 ) / ( 10 - 1 ) ] - n(1) }

= (4/9) [ (10/9) ( 10^n - 1 ) - n ]
= (4/81)[10^(n+1)-9n-10]

Is this solution Helpfull? Yes (15) | No (2)  

KUNWAR PRABHAKAR SINGH  (11 years ago)

putting 1 we will get the 1st term that is 4,so only eqn 2 satisfies thus answer is(3) only

Is this solution Helpfull? Yes (4) | No (3)  

Balakrishnan M  (11 years ago)

4 + 44 + 444 + ... to n terms

= 4 [ 1 + 11 + 111 + ... to n terms ]

= (4/9) [ 9 + 99 + 999 + ... to n terms ]

= (4/9) [ (10 - 1) + (10^2 - 1) + (10^3 - 1) + ... + (10^n - 1) ]

= (4/9) [ ( 10 + 10^2 + 10^3 + ... + 10^n ) - ( 1 + 1 + 1 + ... n times ) ]

= (4/9) { 10 [ ( 10^n - 1 ) / ( 10 - 1 ) ] - n(1) }

= (4/9) [ (10/9) ( 10^n - 1 ) - n ]
= (4/81)[10^(n+1)-9n-10]

Is this solution Helpfull? Yes (3) | No (3)  

Janaki Senthil  (11 years ago)

4+44+444+4444+...=4(1+11+111+1111+ . . .)
Now using geometric series we can evaluate,
1+11+111+1111+...=1+(1*10+1)+{(1*10)*10+1}+ . .
If you replace the 10 with r, it reduces to a geometric series,
1/(r-1)^2 (r^(n+1)-(1+n)r +n)
Hence the result is 1/81{10^(n+1)-10(1+n) +n } =1/81{10^(n+1)-10-10 n+n}
=1/81{10^(n+1)-9 n-10}. . According to our requirement in the question,multiply by 4 . . So the answer is (3) :) ;)

Is this solution Helpfull? Yes (1) | No (2)  

apurva mishra  (11 years ago)

the correct ans. is 3 i.e. (4/81)[10^(n+1)-9n-10]
cause 4(1+11+111+1111+.............)
4/9[(1+11+111+......)*9]
4/9[(10+10^2+10^3+.....)-n]
it's in g.p. now aaplying formula of sum of n terms of gp we get the correct ans. as 3rd option

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