Help with simple proof

[high willow]

I am just dabbling with proofs and need some guidance. Please look at https://math.libretexts.org/Courses/Mount_Royal_University/MATH_1150%3A_Mathematical_Reasoning/3%3A_Number_Patterns/3.1%3A_Proof_by_Induction

For Example 3.1.1:
I understand the following, "We will show that 2^k + 1 > (k + 1) + 4."
What I don't understand is the next line, "Consider 2^k + 1 = 2(2^k) > 2(k + 4) = 2k + 8."
And consequently I don't understand the next line after the one above.

Any insight would be appreciated. Thank you!

[rugged oak]

You mean surely 2^(k+1) not 2^k + 1

[high willow]

yeah

[rugged oak]

Ok

this is out of context what you are asking

You are in the induction step

basically

I assume you proved or understood the base case k = 1?

[high willow]

i get the product rule, but i dont get 2(k + 4) = 2k + 8

where does that come from?

[rugged oak]

"2(k + 4) = 2k + 8" is simply distribution law.

Do you understand mathematical induction at all? Are you familiar with basic mathematical arithmetic?

[high willow]

is the base case k = 3?

[rugged oak]

yes

2^3 = 8
3 + 4 = 7

and 8 > 7

So you can assume for some integer k that is greater or equal to 3 the assumption "2^k > k + 4" works

Next step is to show it for the next (k+1)-th term

[valid sundial]

yummy_pee (𝔸dωn𝓲²s)

[rugged oak]

So they start off with

[valid sundial]

yummy_pee (𝔸dωn𝓲²s)

[rugged oak]

Our assumption is that 2^k is greater than k+4

So we can estimate down the equation

[valid sundial]

yummy_pee (𝔸dωn𝓲²s)

yummy_pee (𝔸dωn𝓲²s)

[rugged oak]

The goal is to derive something truthful

So we can again estimate the inequality down

2k is greater than k+1

if you draw these linear functions

,w plot y = 2x and y = x+1 between 3 and 10

[valid sundial]

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[rugged oak]

So for n >= 3 this estimation down is true

[valid sundial]

yummy_pee (𝔸dωn𝓲²s)

[rugged oak]

and last step to arrive at what we wanted

we can go yet again down

[valid sundial]

yummy_pee (𝔸dωn𝓲²s)

[rugged oak]

that's logical right?

[valid sundial]

yummy_pee (𝔸dωn𝓲²s)

[rugged oak]

Induction is just a matter of setting a goal (what do we need and want to prove and where do we wanna arrive) then step by step try to arrive at where we want to end up.

We started with 2^(k+1) to end up at ... > ... > (k+4) + 1 with our assumption

[mighty berry]

Bro in Induction we basically have to prove that the next number substitution would give us the same form of equation in the form of k where it's a constant

It's an assumption based on the hypothesis that the sequence is valid to make a claim using PMI

Like u say n=1 for a sequence same like here but u take n+1 just to prove you know that it won't make a difference

[high willow]

hey guys. thanks for the help so far. unfortunately, i am still stumped on this part: after "we show that" 2^(k+1) > (k+1)+4, why do we "consider" 2^(k+1)=2(2^k)>2(k+4)=2k+8? Specifically, 2(k+4)=2k+8? How do we go from (k+1)+4 to 2(k+4)=2k+8?

[rugged oak]

bruh

[high willow]

lol i know bro, im a dumbass.

i get the power rule, going from 2^(k+1)=2(2^k)

i can understand that easy enough

its juust confusing me going from (k+1)+4 to 2(k+4)

maybe im looking at it wrong, perhaps im missing something

evidently im not understanding something right at the spot

[high willow]

i think this is where im not understanding

can you describe what you are doing here algebraicly?

@rugged oak

[rugged oak]

I am using our assumptiom

2^k > k+4

which is necessary

[high willow]

ok, how do you get (k+4)(2)?

[rugged oak]

multiply both sides by 2

[high willow]

ok, yeah thats what i was thinking... im not good with algebra, i took it a long time ago in high school so ive been trying to brush up on it

[rugged oak]

if 2^k > k+4 then 2*2^k > (k+4)2 and we can go down

[rugged oak]

Try to read this again step by step

[high willow]

ok so just to clarify, are you performing the product rule here? going from 2^(k+1) to 2(2^k)?

[rugged oak]

yea

you can also start right off with the assumption and multiply both sides by 2

[high willow]

ok, by doing so does that mean you must multiply the other side by 2?

[rugged oak]

yes

[high willow]

obviously i understand that if you multiply both sides by 2 that you get the same equation. i just dont get that by using the product rule that you must also multiply the other side

[rugged oak]

,,2^k > k+4 \implies 2 \cdot 2^k > 2 \cdot (k+4)

[high willow]

ok well yeah i get that. but you were starting with 2^(k+1)

thats where i got confused

[rugged oak]

there are many ways to approach the same problem

[rugged oak]

I am just explaining why this step is correct

because this is true