Considering the next succession, prove by induction that $b_n<0$ for $n \ge 2$ - Is this correct?

Question

Considering the succession

$b_1=2$

$ b_{n+1}= \frac{b_n}{1-b_n}$

Prove by induction that $b_n<0$ for any natural number $n\ge2$

What I've done is: Prove for $n=1$:

$$b_{1+1}=b_2=\frac{b_1}{1-b_1}=\frac{2}{1-2}=-2$$ Which by the assumption made in the exercise, it's correct. Now, to prove that $b_n<0$, I only substituted. Like this:

Knowing that $b_n$ is like this: $$b_{n}=\frac{b_{n-1}}{1-b_{n-1}}$$

For $n=2$: $$b_{2}=\frac{b_{2-1}}{1-b_{2-1}}=\frac{b_{1}}{1-b_{1}}=\frac{2}{1-2}=-2$$

Which confirms the theory exposed in the exercise above.

Just in case, I tried it with $n=3$ and the result still came up with a negative value. $$b_{3}=\frac{b_{3-1}}{1-b_{3-1}}=\frac{b_{2}}{1-b_{2}}=\frac{-2}{1+2}=\frac{-2}{3}$$

Is this enough to prove the exercise? Or is it all wrong? Thanks for the help!

The exercise also asks if the succession $b_n$ grows. What I done was subtract $b_{n+1}-b_{n}>0$ (bigger than zero so it grows) Is it ok?

Answer 1

You need to prove the statment for every $n$, not just for $n=1$, $n=2$ and $n=3$. So the natural question is how to prove some statment for every natural number (greater than two) when there is infinetly many natural numbers? How to check the statment for infinetly many cases?

Solution is to use mathematical induction. Let us say that you have some statement that depends on $n$ (like one you have in your question). Induction demands $3$ steps.

Step #1: Base of induction. You need to check your statement for one natural number. So let us check your statement for $n=2$ (because you need to check it for every $n$ greater than two): $ b_{2}=\dfrac{b_{1}}{1-b_{1}}= \dfrac{2}{1-2}=-2<0.$

Step #2: Assumption of induction. In this step we make an assumption. We assume that your statement works for some arbitrary number $k$. That means we assume that for some arbitrary number $k$: $b_{k}<0$.

Step #3: Inductive step. Now, we proove that then it must work for $k+1$. So here it is: $ b_{k}<0 \implies \dfrac{b_{k}}{1-b_{k}} <0 $, this is because $b_{k}$ is negative. So $1-b_{k}$ will be positive so when you divide negative ($b_{k}$) and positive($1-b_{k}$) you get again negative (that means less than zero). But that means $b_{k+1}$ must be negative because: $0>\dfrac{b_{k}}{1-b_{k}}=b_{k+1}$. Now what we have proved is that if yor statement works for some arbitrary $k$ that it works for $k+1$. But your statement works for number chosen in base of induction, namely number 2. That means that it has to work for number $2+1=3$. But now when it works for number $3$ it must work for $3+1=4$ and so it will work for every natural number.

So I will now solve second part when you need to see that sequance gets bigger. Sequance gets bigger only after $n\geq 2$.

Step #1. $b_{2}=-2, b_{3}=-\dfrac{2}{3} \implies b_{3}>b_{2}.$

Step #2. We make assumption that for some arbitrary $k$ statement $b_{k+1}>b_{k}$ is true.

Step #3. Let us first note: $b_{k+1}=\dfrac{b_{k}}{1-b_{k}} \implies \dfrac{1}{b_{k+1}}=\dfrac{1-b_{k}}{b_{k}}=\dfrac{1}{b_{k}}-1.$ Now we have: $b_{k+1}>b_{k} \implies \dfrac{1}{b_{k+1}}<\dfrac{1}{b_{k}} \implies \dfrac{1}{b_{k+1}}-1<\dfrac{1}{b_{k}}-1 \implies \dfrac{1}{b_{k+2}}<\dfrac{1}{b_{k+1}} \implies b_{k+2}>b_{k+1}.$

And then by mathematical induction it is proved that for every $n>2$ sequence get bigger.

Answer 2

Hint: $\;\dfrac{1}{b_{n+1}} = \dfrac{1-b_n}{b_n} = \dfrac{1}{b_{n}} - 1\,$, so if $\,b_n\,$ is negative then $\,\ldots$

Also, $\;\dfrac{1}{b_{n+1}} = \dfrac{1}{b_{n}} - 1 \lt \dfrac{1}{b_n}\,$, so $\,\dfrac{1}{b_n}\,$ is decreasing, and therefore $\,b_n\,$ is $\;\ldots$

Answer 3

You assume that for one particular $n$ that you now that $b_n< 0$. So now you have to prove that $\frac {b_n}{1 - b_n} < 0$. Now $b_n < 0$ so that means $\frac {b_n}{1-b_n} < 0$ if and only if $1 - b_n > 0$. And that is true if and only if $b_n < 1$. Which... it is... because $b_n < 0$.

So the proof is:

Proposition: For any $n \ge 2$ $b_n < 0$.

Proof of $n = 2$. $b_2 = \frac {b_1}{1-b_1} = \frac 2{1-2} -2 < 0$ so it is true for $n= 2.

If it is true for $n=k$ then here is the proof that it is true for $n=k+1$.

$b_k < 0$ so $1-b_k > 1 > 0$. And $b_k < 0$ so $b_{k+1} = \frac {b_k}{1-b_k} < 0$.

That's it. Intitial step. Proposition is true for $n=2$. Inductive step. If true for any $n = k$ it will be true of $n =k+1$. So it is true for all $n \ge 2$.