check if inequality is real

Question

check if inequality is real:

30n < 2^n + 105

my solution proposal:

1. n=1 30<107 true
2. n=k 30k < 2^k +105

30(k+1) < 2^{k+1} + 105 Proof 30(k+1) < 2^{k+1} + 105

30k+30< 2^k +105+30=2^k +135

2^k +135 < 2*2^k+105

2^k + 135 < 2^k + 2^k +105

30 < 2^k

K> 5

n=5 30(5+1)< 2^{5+1} +105 180< 2^6+105=169 this is not true...

So, which conclusion follows from this ? what can you say about it? I have no conclusions about this inequality ... Help

Answer 1

I would write $$2^n>30n-105$$ then we have to prove that $$2^{n+1}>30(n+1)-105$$ Multiplying $$2^n>30n-105$$ by $2$ then $$2^{n+1}>60n-210$$ Now you have to prove that $$60n-210>30(n+1)-105$$