MHB Inequality proof w/ induction, 2 unknowns

AI Thread Summary
The discussion centers on proving the inequality \((1+x)^n \geq 1 + nx\) for real \(x > -1\) and natural numbers \(n \geq 2\) using mathematical induction. The base case for \(n=2\) is established as true, showing \((1+x)^2 \geq 1 + 2x\). The inductive step involves assuming the statement holds for \(n=m\) and proving it for \(n=m+1\). The proof is completed by manipulating the expression \((1+x)^{m+1}\) and applying the inductive hypothesis, ultimately confirming that the inequality holds. The participants successfully clarify the steps needed to complete the proof, emphasizing the importance of constructing inequalities during the induction process.
skate_nerd
Messages
174
Reaction score
0
I am given a statement to prove: Show (without using the Binomial Theorem) that \((1+x)^n\geq{1+nx}\) for every real number \(x>-1\) and natural numbers \(n\geq{2}\). I am given a hint to fix \(x\) and apply induction on \(n\).
I started by supposing \(x\) is a fixed, real number larger than -1, and then calling the given formula \(P(n)\), and evaluating \(P(n)\) at the base case \(n=2\).
This gives \((1+x)^2\geq{1+2x}\) which can be rewritten as \(1+2x+x^2\geq{1+2x}\).
It is know that for all real \(x\), the statement \(x^2\geq{0}\) is true.

Here is where I get tripped up.
We need to assume that \(m=n\) a.k.a. \(P(m)\) is true for all natural \(m\geq{2}\).
So we have \((1+x)^m\geq{1+mx}\). Now we need to show that \(P(m+1)\) holds to be true. \(P(m+1)\):
\((1+x)^{m+1}\geq{1+(m+1)x}\).
Now here I would usually try to translate the original formula by plugging in what we had originally, but I am pretty iffy on how to do this with an inequality. If anybody could help me construct a new formula that would help me prove that
\((1+x)^{m+1}\geq{1+(m+1)x}\) is true I would be very thankful.
 
Physics news on Phys.org
skatenerd said:
I am given a statement to prove: Show (without using the Binomial Theorem) that \((1+x)^n\geq{1+nx}\) for every real number \(x>-1\) and natural numbers \(n\geq{2}\). I am given a hint to fix \(x\) and apply induction on \(n\).
I started by supposing \(x\) is a fixed, real number larger than -1, and then calling the given formula \(P(n)\), and evaluating \(P(n)\) at the base case \(n=2\).
This gives \((1+x)^2\geq{1+2x}\) which can be rewritten as \(1+2x+x^2\geq{1+2x}\).
It is know that for all real \(x\), the statement \(x^2\geq{0}\) is true.

Here is where I get tripped up.
We need to assume that \(m=n\) a.k.a. \(P(m)\) is true for all natural \(m\geq{2}\).
So we have \((1+x)^m\geq{1+mx}\). Now we need to show that \(P(m+1)\) holds to be true. \(P(m+1)\):
\((1+x)^{m+1}\geq{1+(m+1)x}\).
Now here I would usually try to translate the original formula by plugging in what we had originally, but I am pretty iffy on how to do this with an inequality. If anybody could help me construct a new formula that would help me prove that
\((1+x)^{m+1}\geq{1+(m+1)x}\) is true I would be very thankful.

\displaystyle \begin{align*} \left( 1 + x \right) ^{m + 1} &= \left( 1 + x \right) \left( 1 + x \right) ^m \\ &\geq \left( 1 + x \right) \left( 1 + m\,x \right) \\ &= 1 + \left( m + 1 \right) \, x + m\,x^2 \\ &\geq 1 + \left( m + 1 \right) \, x \end{align*}
 
Okay, you have shown the base case is true, and so your induction hypothesis $P_m$ is:

$$(1+x)^m\ge1+mx$$

I would consider for the inductive step:

$$(1+x)^{m+1}-(1+x)^{m}=(1+x)^{m}(1+x-1)=(1+x)^{m}x$$

Can you use the inductive hypothesis to construct from this a weak inequality that you can then add to $P_m$?
 
Thanks to both of you for the responses. So I was actually able to figure out and finish the proof going by what Prove It wrote. But to MarkFL, what do you mean by constructing a "weak inequality"? Does that refer to the point where you find an inequality where you have \(mx^2\geq{0}\)?
 
What I had in mind is to take the equation:

$$(1+x)^{m+1}-(1+x)^{m}=(1+x)^{m}x$$

and then use the induction hypothesis (which we have multiplied by $x$) to write:

$$(1+x)^{m}x\ge(1+mx)x$$

so that we now have:

$$(1+x)^{m+1}-(1+x)^{m}\ge(1+mx)x$$

Adding this to $P_m$, we get:

$$(1+x)^{m+1}\ge 1+mx+(1+mx)x=1+(m+1)x+mx^2$$

Since $mx^2\ge0$, we have:

$$(1+x)^{m+1}\ge1+(m+1)x+mx^2\ge1+(m+1)x$$

And so we may conclude:

$$(1+x)^{m+1}\ge1+(m+1)x$$

Thus, we have derived $P_{m+1}$ from $P_{m}$, thereby completing the proof by induction.
 
Ahh I see. Yeah I ended up writing something similar to that, with the same ending. Thanks for the help guys!
 
Back
Top