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51.

Prove the following by using the principle of mathematical induction for all `n in N`:`(2n+7)

Answer» Check by putting the value of n=1,2,3,4 you get the result.
Discussion
52.

For each `n in N, n(n+1) (2n+1)` is divisible byA. 15B. 8C. 7D. 6

Answer» Correct Answer - D
Discussion
53.

Choose the propostion among the following, that is not true for n=1 but true for all `n gt 1`.A. `n! gt ((n+1)/(2))^(n)`B. `n! gt ((n+1)/(2))^(n)` is an integerC. `(d^(n))/(d xx^(n))(sin ax+b)=a^(n)sin(ax+b+(npi)/(2))`D. `n! lt ((n)/(2))^(n)`

Answer» Correct Answer - A
Discussion
54.

Choose the proposition among the following that is not true for all `n in N`A. `2^(n) gt n`B. `2^(n)gt n^(2)`C. `1+2+3+…+n lt (2n+1)^(2)`D. `(2n+7) lt (n+3)^(2)`

Answer» Correct Answer - B
Discussion
55.

The statement `2^(n) ge n^(2)` (where `n in N)` is true forA. All nB. `n ge 3`C. `n lt 4`D. `n gt 4`

Answer» Correct Answer - D
Discussion
56.

Choose the statement which is correct for all `n in N`A. `2n gt 2n+1`B. `x^(n)-y^(n)` is divisible by (x+y)C. `x^(n)-y^(n)` is divisible by (x-y)D. `1+(1)/(4)+(1)/(9)+(1)/(16)+....+(1)/(n^(2)) lt 2-(1)/(n)`

Answer» Correct Answer - C
Discussion
57.

n(n+1)(n+2) is divisible by k for `AA n in N`. The largest k isA. 2B. 3C. 6D. 12

Answer» Correct Answer - C
Discussion
58.

`x^(n)+y^(n)` is divisible by (x+y) is true when `n(in N)` is of the form `(k in N)`.A. 4k+1B. 4k+3C. 4k+7D. 2k

Answer» Correct Answer - D
Discussion
59.

Prove the following by using the principle of mathematical induction. n(n+1)+1 is an odd natural number, `n in N`.

Answer» Check for P(1), P(2)….P(6)
we get smallest positive integer 6.
Discussion
60.

Prove that `tan^(- 1)(1/3)+tan^(- 1)(1/7)+tan^(- 1)(1/13)+..........+tan^-1 (1/(n^2+n+1))+......oo =pi/4`A. `tan^(-1)((n)/(n+2))`B. `tan^(-1)((2n)/(n+2))`C. `tan^(-1)((3n)/(n+2))`D. `tan^(-1)((n)/(n+4))`

Answer» Correct Answer - A
Discussion
61.

Statement-1: `1+2+3....+n=(n(n+1))/(2),"for all "n in N` Statement-2: `a+(a+d)+(a+2d)+....+(a+(n-1)d)=(n)/(2)[2a+(n-1)d]`

Answer» Correct Answer - A
Discussion
62.

The sum of the square of three consecutive odd number increased by 1 is divisible by (use mathematical indcution).A. 12 as well as 24B. 12 but not 24C. Neither by 12 nor by 24D. By all multiples of 12

Answer» Correct Answer - B
Discussion
63.

Statement-1: `cos((pi)/(10))cos((2pi)/(10))cos((4pi)/(10))=(cos((8pi)/(10)))/(2^(3)sin((pi)/(10)))` Statement-2: `cos A.cos2A.cos2^(2)A.......cos(2^(n-1)A)=(sin(2^(n)A))/(2^(n)sinA)`

Answer» Correct Answer - C
Discussion
64.

Most of the formulae are verified on the basis of induction method by putting n=1,2,…. Which of the following is correct?A. `2^(n) gt n!`B. `1!+2.2!+3.3!+...+n.n=(n+1)!-1`C. `1+2+3...+n=(n(n-1))/(2)`D. `1^(2)+2^(2)+...+n^(2)=(n(n+1)(2n+1))/(3)`

Answer» Correct Answer - B
Discussion