Question

help me prove by induction, please
2n + 1 ≤ 2^n
, for all integers n ≥ 3.

Answer 1

For sure this is true for \(n = 3\) because \(7=2*3+1 \le 3^3=8\)
So assume \(2n+1\le 2^n \) for some \(n\)
We need to show that \(2(n+1)+1\le n^{ n+1}\)
So \(2(n+1)+1=(2n+1)+2\) and \(2^{n+1}=2*2^n=2^n+2^n\)
We know by assumption \(2n+1\le 2^n\) and for sure \(2\le2^n \ \ \forall n\in \mathbb{N} \)
So \((2n+1)+2=2(n+1)+1\le 2*2^n=2^{n+1}\), thus by induction \[2n+1\le 2^n\text{ for all }n\ge3 \ \ \ \ \ \ \ \ _\square\]

Answer 2

note: the \(n\ge3\) part is for the first step.

Answer 3

@academicpanda understand?

Answer 4

Yes, you do it better than the solution. look

Answer 5

that first line should say \(\le 2^3=8\)....

Answer 6

and \(\le 2^{n+1}\) on the third line.

Answer 7

they make it so confusing

Answer 8

yeah I can explain that but I dont think its the best way.

Answer 9

So (2n+1)+2=2(n+1)+1≤2∗2n=2n+1, thus by induction

Answer 10

should it not be 2(n+1)-1 ?

Answer 11

2(n+1)+1 = 2n+2+1 = 2n+3 = (2n+1)+2

Answer 12

oh sorry never mind., I see it. My eyes are so blurry. I am super tired.

Answer 13

word

Answer 14

could I ask you for your help two more please?

Answer 15

you explain very straight forward. Love it

Answer 16

sure, I would close this and open another

Answer 17

ok :)

Answer 18

I will post it