Question

Prove that \[7n+3\le2^{n}\] for all \[n \ge6\]

Answer 1

Proof by induction!

Answer 2

Let's see if the relation is valid for \(n=6\),
\[7(6)+3\le 2^6 \implies 45\le 64\] TRUE!
Now, assume it's true for n=k, we have to prove it's also true for n=k+1.

Answer 3

Prove for n = 6:

42 + 3 <= 2^6
45<=64

True!

Suppose it's true for n:

7n + 3 <= 2^n

prove for n+1

7(n+1) + 3 <= 2^(n+1)

7n + 3 + 7 <= 2 * 2^n

7n + 3 + 7 <= 2^n + 2^n

(7n + 3) + 7 <= 2^n + 2^n

Now we said suppose it's true for n:

therefore 7n + 3<= 2^n

Now we must prove 7<=2^n

Since we know it's true for n>=6:

2^6 > 7

Therefore 7<=2^n is true for n>=6

that proves it for n+1

Therefore it's always true

Answer 4

well, not always. true for n>=6

Answer 5

2 * 2^n = 2^n + 2^n ?

Answer 6

You didnt factor anything there right

Answer 7

its just a basic arithmetic fact

Answer 8

like 3*2^n would be 2^n + 2^n + 2^n

Answer 9

Damn it! I got confused! I need a paper :P

Answer 10

You can factor it, yes. 2*2^n = (1 + 1) * 2^n = 2^n + 2^n

Answer 11

ah ok

Answer 12

Mr Math, check if my proof makes sense please? I've never really done an inequality induction before

Answer 13

Yeah, slaaibak's work is correct.

Answer 14

I dont see how its proof

Answer 15

http://en.wikipedia.org/wiki/Mathematical_induction

basically if you prove for the base case, then thereafter if you prove for n+1, you prove for all natural numbers. since you first proof for 6, then 6+1=7, then 7+1 and it goes on forever

Answer 16

Right

Answer 17

I just dont get it lol

Answer 18

so wait

Answer 19

We assumed \(7k+3\le 2^k\), we want to prove it for n=k+1, that's
\[(7k+3)+7\le 2^k+2^k \text{, but we have } 7k+3\le 2^k \implies 7\le 2^k\]. \[\text{ This is true for }k\ge3.\]

Answer 20

we assume 7n + 3 <= 2^n is true

Answer 21

And hence the relation is true for all integers \(k\ge 6\).

Answer 22

@mwmnj, You probably need to look up proof by induction and "its proof". :D

Answer 23

Is that it?

Answer 24

Cause there isnt an intuitive obvious proof im seeing when i do these

Answer 25

Its just a pattern im following

Answer 26

Mathematical induction is one of the greatest methods of proofs. http://en.wikipedia.org/wiki/Mathematical_induction