Prove that `(1+x)^ngeq(1+n x),`for all natural number n, where `x >-1.`

Prove that (1+x)^(n)>=(1+nx) for all natural number n,where x>-1

Prove that 2n +1 lt 2^(n) for all natural numbers n ge3

Using the principle of mathematical induction prove that (1+x)^(n)>=(1+nx) for all n in N, where x>-1

For a gt 01 , prove thjat (1+a)^(n) ge (1+an) for all natural numbers n.

prove that 1+5+9+ . . .+(4n-3)=n(2n-1), for all natural number n.

Prove that log_(n)(n+1)>log_(n+1)(n+2) for any natural number n>1

Prove that 1+2+2^(2)+ . . .+2^(n)=2^(n+1)-1 , for all natural number n.

Let x_(n)=(2^(n)+3^(n))^(1//2n) for all natural number n. Then

The natural number a for which sum_(k=1)^(n)f(a+k)=2n(33+n) where the function f satisfies the relation f(x+y)=f(x)+f(y) for all the natural numbers x,y and further f(1)=4 is

Find the natural number a for which sum_(k=1)^(n)f(a+k)=16(2^(n)-1), where the function f satisfies the relation f(x+y)=f(x)f(y) for all natural number x,y and,further,f(1)=2.