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1+ ((1)/(3) + (1)/(3^2) ) + (( 1)/( 3^3)...

`1+ ((1)/(3) + (1)/(3^2) ) + (( 1)/( 3^3) + (1)/( 3^4) + (1)/( 3^5) ) + ....` sum of the terms in the `n^( th)` bracket=

A

`((3n-1)^(3) )/( 2.4^(n-1) )`

B

`(3n-1)/( 2.3^( (n-1) (n+2) //2) )`

C

`(3^n +1)/( 3.7^(n-1) ) `

D

`(3^(n -1) )/( 2^(n+1) )`

Text Solution

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The correct Answer is:
B
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(1)/(3)+(1)/(2.3^(2))+(1)/(3.3^(3))+(1)/(4.3^(4))+….oo=

(1)/(1.3) + (1)/(3.5) + (1)/(5.7) + …. (n-3) terms

1 + 1/3 + (1.3)/(1.2) .(1)/(3^2) + (1.3.5.)/(1.2.3).(1)/(3^3) + ……oo =

2[((1)/(3))+(1)/(3)((1)/(3))^(3)+(1)/(5)((1)/(3))^(5)+...] =

e^(2((1)/(3)+(1)/(3)*(1)/(3^(3))+(1)/(5)*(1)/(3^(5))+….))=

Sum of n terms of (1)/(1.2) + (1)/(2.3) + (1)/(3.4) + …. is

(1)/(5)+(1)/(2.5^(2))+(1)/(3.5^(3))+(1)/(4.5^(4))+.......=

(1)/(1.3).(1)/(2)+(1)/(2.4).(1)/(2^(2))+(1)/(3.5).(1)/(2^(3))+....=

(1)/(2n^(2)-1)+(1)/(3(2n^(2)-1)^(3))+(1)/(5(2n^(2)-1)^(5))+....=

The sum (3)/(1^(2)) + (5)/(1^(2) + 2^(2)) + (7)/(1^(2)+ 2^(2) + 3^(2)) + ….. upto 11 terms is

AAKASH SERIES-MATHEMATICAL INDUCTION-PRACTICE SHEET (EXERCISE-I) LEVEL-I (Principle of Mathematical Induction) (Straight Objective Type Questions)
  1. If sum (r=1)^(n) t(r) = (1)/( 12) n (n+1) (n+2) then sum(r=1)^(n) (1)...

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  2. 1+ (1+3) + (1+3+ 5) + ….n brackets =

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  3. 1+ ((1)/(3) + (1)/(3^2) ) + (( 1)/( 3^3) + (1)/( 3^4) + (1)/( 3^5) ) ...

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  4. Sum of n^( th) bracket of (1) + (2+ 3+4) + ( 5+ 6+ 7+ 8+9) + ….. is

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  5. { n (n+1) (2n+1) : n in Z} sub

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  6. If a,b and n are natural numbers then a^(2n-1) + b^(2n-1) is divisible...

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  7. If 'a' and 'b'are natural numbers such that a^(2) - b^(2) is prime num...

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  8. AA n in N, 49^(n) + 16n -1 is divisible by

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  9. AA n in , 7^(2n) + 3^(n-1) . 2^(3n-3) is divisible by

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  10. Sum of the cubes of three successive natural number is divisible by

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  11. For every natural number n, 3^(2n+2) - 8n -9 is divisible by

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  12. If 10^(n) + 3.4^(n) + x is divisible by 9 for all n in N, then least ...

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  13. AA n in N, 5^(2n+2) - 24n - 25 is divisible by

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  14. If y^(n) -1 is divisible by y+ a for all values of n in N, then a must...

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  15. 7^(13) + 6^(13) is divisible by

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  16. (1+ x)^(n) -nx -1 is divisible by (where n in N )

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  17. For a positive integer (n gt 1), Let a (n) = 1+ (1)/(2) + (1)/(3) + (1...

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  18. For all positive integers n gt 1, { x(x^(n-1) - n,a^(n-1) ) } + a^(n) ...

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  19. ((n+2)!)/( (n-1)!) is divisible by

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  20. When 2^( 301) is divided by 5, the least +ve remainder is

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