Complete the following statements.
and `b_n=1-a_n` then the smallest natural number `n_0`, such that `b_n > a_n Vnt n> n_0` is

If n is a natural number then show that n! + (n+1)! = (n+2)n!

For natural number n , 2^n (n-1)!lt n^n , if

If A_n = P_n + 1 , where P_n is the product of the first n prime numbers, then consider the following statements : 1. A_n is always a composite number. 2. A_n + 2 is always an odd number. 3. A_n +1 is always an even number. Which of the above statements is/are correct ?

Consider the following two statements : I. if n is a composite number, then n divides (n – 1)! II. There are infinitely many natural numbers n such that n^3 + 2n^2 + n divides n!. Then

Let A_n=(3/4)-(3/4)^2+(3/4)^3+….+(-1)^(n-1)(3/4)^n and B_n = 1-A_n . find the least odd natural numbers n_0 , so that B_ngtA_n A for all ngen_0

If n is the smallest natural number such that n+2n+3n+...+99n is a perfect square, then the number of digits in n^(2) is

Consider the following two statements : I. If n is a composite number, then n divides (n - 1)!. II. There are infinitely many natural numbers n such that n^(3)+2n^(2)+n divides n!. Then