Question

using Mathematical Induction prove that for any natural number N the statement 4 to the power 2n is greater than 15n and is always true

Answer 1

Solution:

The given statement is : $4^{2 n} > 15^n$

1. For n= 1,

L.H.S= $4^2=16$

R.H.S= $15^1$= 15

Suppose this statement is true for , n=k, that is

     $4^{2 k} > 15^k$------(1)

Now we will prove that this statement is true for , n= k +1

$4^{2 (k+1)} > 15^(k+1)$

L.H.S= $4^{2(k+1)}=4 ^{2 k} . 4^2>15^k \times (15 +1) \\\\ 4^{2k}.4^2>15 ^k \times 15 + 15 ^k\\\\ 4^{2 k+2}> 15^{k+1} + 15 ^k \\\\  4^{2 (k+1)} > 15^{k+1}$---Using (1)

Hence proved

Answer 2

Answer:

Step-by-step explanation:

to prove

$4^{2n}> 15n$

we can write it as

$16^{n}> 15n$

by taking minimum value of (n=0) equation become

⇒1> 0

if we take value of n=1, equation become

⇒16> 15

as we increase the value of n, 16^{n} will always greater than 15n

so it clearly show that $4^{2n}> 15n$ will always true for any value of n.