Related InterviewSolutions

• Let `I_(m)=int_(0)^(pi)((1-cos mx)/(1-cos x))dx` use mathematical induction to prove that `l_(m)= m pi, m=0,1,2`......
• Prove by mathematical induction that `(1)/(1+x)+(2)/(1+x^2)+(4)/(1+x^4)+.....+(2^n)/(1+x^(2^n))=(1)/(x-1)+(2^(n+1))/(1-x^(2^(n+1)))` where , `|x|ne 1 ` and n is non - negative integer.
• Using the principle of mathematical induction to prove that `int_(0)^(pi//2)(sin^2nx)/(sinx)dx=1+(1)/(3)+(1)/(5)+.....+(1)/(2n-1)`
• `n^7-n` is divisible by 42 .
• Prove by induction that the integer next greater than `(3+sqrt(5))^n` is divisible by `2^n` for all `n in N`.
• For all `n in N, Sigma n`A. `lt((2n+1)^(2))/(8)`B. `gt((2n+1)^(2))/(8)`C. `=((2n+1)^(2))/(8)`D. none of these
• For all `n in N, 1 + 1/(sqrt(2))+1/(sqrt(3))+1/(sqrt(4))++1/(sqrt(n))`A. equal to `sqrt(n)`B. less than or equal to `sqrt(n)`C. greater tha or equal to `sqrt(n)`D. none of these
• Statement -1 for all natural numbers n , `2.7^(n)+3.5^(n)-5` is divisible by 24. Statement -2 if f(x) is divisible by x, then `f(x+1)-f(x)` is divisible by `x+1,forall x in N`.A. Statement -1 is true , Statement -2 is true, Statement -2 is correct explanation for Statement -2B. Statement -1 is true , Statement -2 is true , Statement -2 is not correct explanation for Staement -2C. Statement -1 is true , Statement -2 is falseD. Statement -1 is false , Statement - 2 is true.
• `AA n in N, 49^n+16n-1` is divisible by (A) `64` (B) `49` (C) `132` (D) `32`A. 64B. 8C. 16D. 4
• If n is an odd positive integer, then `a^(n)+b^(n)` is divisible byA. a+bB. a-bC. `a^(2)+b^(2)`D. none of these