Let n with n 2 Prove by induction that 3n n 2n Hint If n 2 t

Let n with n 2. Prove by induction that 3ⁿ n 2ⁿ. Hint: If n 2, then 3n 2n + 2 (why?).

n is any natural number greater than or equal to 2.

n can be 2,3,4,5......

To prove that 3n>=2n+2

Let n =2 then 3n = 6 >=2(2)+2

Thus true for n =2

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Assume true for n = k

i.e. 3k >= 2k+2

To prove that true for n = k+1

To prove that 3(k+1) >=2(k+1)+2

Left side = 3k+3 >= 2k+2+3 by the assumption for n = k

i.e. Left side >= 2k+5 = 2k+4+1

=2(k+1) +3

>= 2(k+1)+2

Thus if true for n=k true for n = k+1 also

Holds good for all natural numbers