okay I know how to solve these. But here is the problem i\'m working on and I ne

Question

okay I know how to solve these. But here is the problem i'm working on and I need help understanding the algebra or something because it has been so long.
n^2 < 2^n, for all integers n >= 5

when I get to the inductive step and am proving p(k+1) i've marked the area I don't understand in red. Please help me understand this.

p(k+1) : (k+1)^2 =
= k^2 +2k +1
< 2^k + 2k + 1 : Inductive hypothesis
< 2^k + 2^k                      :right here I dont understand how 2k+1 = 2^k
= 2 * 2^k                          :right here I dont understan how 2^k +2^k = 2 *2^k
= 2^(k+1)

Explanation / Answer

for the first line we want:
2k+1<k^2

k^2-2k+1>2
(k-1)^2>2

k>1+2

so for k>=5>1+2, we have 2k+1<k^2 <2^k

the second part is indeed so easy. for any number n we have:

n+n=n(1+1)=2n

now replace n by 2^k

2^k+2^k= 2^k (1+1)= 2 * 2^k

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