Mathematical induction problem

[DDarthVader]

Hello! First of all I have like 5 exercises I don't quite understand so will it be a problem if I create 5 new topics in the next 24h?

Homework Statement

Prove, by using mathematical induction that if x+1 \geq 0 then (1+x)^n \geq 1+nx.

Homework Equations

The Attempt at a Solution

Basic step: If n=1 then 1+x \geq 1+x which is true.
Induction Step: Now making n=k we get (1+x)^k \geq 1+kx. If the hypothesis holds for n=k then it will hold for n=k+1. Making n=k+1 we get (1+x)^k + (1+x)^{k+1} \geq 1+(k+1)x. And from this:
(1+x)^k(1+(1+x)) \geq 1+(k+1)x
But by our induction hypothesis (1+x)^k \geq 1+kx which means that (1+x)^k(1+(1+x)) \geq 1+(k+1)x is true.

Thanks!

 

[SammyS]

Hello! First of all I have like 5 exercises I don't quite understand so will it be a problem if I create 5 new topics in the next 24h?

Homework Statement

Prove, by using mathematical induction that if x+1 \geq 0 then (1+x)^n \geq 1+nx.

Homework Equations

The Attempt at a Solution

Basic step: If n=1 then 1+x \geq 1+x which is true.
Induction Step: Now making n=k we get (1+x)^k \geq 1+kx. If the hypothesis holds for n=k then it will hold for n=k+1. Making n=k+1 we get
This looks like a typo. → (1+x)^k + (1+x)^{k+1} \geq 1+(k+1)x.
And from this:
(1+x)^k(1+(1+x)) \geq 1+(k+1)x
But by our induction hypothesis (1+x)^k \geq 1+kx which means that (1+x)^k(1+(1+x)) \geq 1+(k+1)x is true.

Thanks!

The induction step is:

Assume the following is true: (1+x)^k \geq 1+kx\,.

From that you need to show that the following is true: (1+x)^{k+1} \geq 1+(k+1)x\,.​

It looks to me as if you're assuming the hypothesis holds for n = k+1 .

 

[DDarthVader]

The induction step is:

Assume the following is true: (1+x)^k \geq 1+kx\,.

From that you need to show that the following is true: (1+x)^{k+1} \geq 1+(k+1)x\,.​

It looks to me as if you're assuming the hypothesis holds for n = k+1 .

That typo is actually not a typo. And I'm trying to say that if the $n=k$ holds then I'll try to prove that $n=k+1$ also holds by doing $(1+x)^k + (1^x)^{k+1}$. I can write it clearer in my language. But the main problem here is $(1+x)^k + (1+x)^{k+1}$. Is this correct?

 

[SammyS]

That typo is actually not a typo. And I'm trying to say that if the $n=k$ holds then I'll try to prove that $n=k+1$ also holds by doing $(1+x)^k + (1^x)^{k+1}$. I can write it clearer in my language.

No, the following is not correct.

But the main problem here is $(1+x)^k + (1+x)^{k+1}$. Is this correct?

If you substitute k+1 for n in \displaystyle (1+x)^n \geq 1+nx\,, then you get \displaystyle (1+x)^{k+1} \geq 1+(k+1)x\,.

By the Way: \displaystyle (1+x)^{k+1}=(1+x)^k\cdot(1+x)\,, it's not the same as (1+x)^k+(1+x)^{k+1}\,.

Maybe you're thinking of \displaystyle \sum_{n=0}^{k+1}(1+x)^n=\left(\sum_{n=0}^{k}(1+x)^n\right)+(1+x)^{k+1}\,.