Principle of Ma...

Let P(n):n² − n + 41 is a prime number . then :

Let that P(n) ⇒ P(n + 1) for all natural numbers n. also , if P ( m ) is true , m ∈ N , then we conclude that

Consider the statement P ( n ) : “n² ≥ 100 “ . Here P(n) ⇒ P(n + 1) for all natural numbers n . Does it mean

The smallest positive integer ‘ n ‘ for which P ( n ) : 2ⁿ<(1×2×3×............×n) holds is :

If xⁿ−1 is divisible by x – k for all n belongs to natural numbers N , then the least positive integral value of k is :

A student was asked to prove a statement P ( n ) by method of induction . He proved P ( k + 1 ) is true whenever P ( k ) Is true for all k ≥ 5 , k ∈ N and P ( 5 ) is true . On the basis of this he could conclude that P ( n ) is true

If a,b,c ∈ N,aⁿ + bⁿ is divisible by c , when n is odd but not when n is even , then the value of c is :

1.2.3 + 2.3.4 + 3.4.5 + ………..up to n terms is equal to 1/4 n ( n + 1 ) ( n + 2 ) ( n + 3 ) is true for 

1 + 2  + 3 + .................. n = 1/2(n(n + 1)), is trure