Consider the statement
`P(n):9^(n)-1` is a multiple of 8, where n is a natural number
i. Is P(1) true?
ii. Assuming P(k) is true, show that P(k + 1) is also true.

Consider the statement : " P(n) : n^(2)-n+41 is prime." Then, which one of the following is true?

Let P(n) denote the statement that n^(2)+n is odd. It is seen that P(n) = P(n + 1). P(n) is true for all.

The inequality n!gt2^(n-1) is true for

Let P(n) be the statement: " n^(3)+n is divisible by 3." i. Prove that P(3) is true. ii. Verify whether the statement is true for n = 4.

Each of the statement p implies~q,~rimplies q and p is true. Then

Consider a G.P. with first term (1+x)^(n) , |x| lt 1 , common ratio (1+x)/(2) and number of terms (n+1) . Let 'S' be sum of all the terms of the G.P. , then sum_(r=0)^(n)"^(n+r)C_(r )((1)/(2))^(r ) equals

Assertion: 1+2 + 3 + ... n = (n(n+1))/2 Reason: In a statement P(n), if P(l) is true and assuming P(k) and if we prove P(k + 1) is also true then P(n) is true for all value n, n is true integer

For the following question, choose the correct answer from the codes (a), (b), (c) and (d) defined as follows: Statement I is true, Statement II is also true; Statement II is the correct explanation of Statement I. Statement I is true, Statement II is also true; Statement II is not the correct explanation of Statement I. Statement I is true; Statement II is false Statement I is false; Statement II is true. Let a , b , c , p , q be the real numbers. Suppose alpha,beta are the roots of the equation x^2+2p x+q=0 and alpha,beta/2 are the roots of the equation a x^2+2b x+c=0, where beta^2 !in {-1,0,1}dot Statement I (p^2-q)(b^2-a c)geq0 and Statement II b !in p a or c !in q adot

Let n be a natural number. Then the range of the function f(n)=""^(8-n)P_(n-4),4lenle6 , is

whether the statement is true or false: n_(1), n_(2),...,n_(p) are p positive integers, whose sum is an even number, then the number of odd integers among them is odd.