# Sequence of b_{k} with Explicit Formula: Proving by Math Induction

• MHB
• hkhgg
In summary, the given conversation discusses a sequence defined by $b_k=\frac{b_{k-1}}{2+b_{k-1}}$ with an initial value of $b_0=1$. The formula for the sequence was calculated to be $1, \frac{1}{3}, \frac{1}{7}, \frac{1}{15}, ... , \frac{1}{2^{n+1}-1}, ...$. However, it is unclear if this is the intended formula as the conversation also mentions a possible alternative formula $b_k=\frac{b_{k-1}}{2}+b_k=\left(\frac{3}{2}\right)b_{k-1}$.
hkhgg
$$\displaystyle b_{k} = b_{k - 1}/2 +b_{k-1}$$
$$\displaystyle b_{0} = 1$$

What would be the sequence for this expression, I calculated it to be 1, 2/3, 2/5, 2/7 ...

Is it right?

My explicit formula is $$\displaystyle b_{n} = 2/n+2$$

What would be the explicit formula in your view and how can that formula be proved by mathematical induction? thanks

First, what is the formula defining the sequence? You wrote $b_k= b_{k-1}/2+ b_{k-1}$ but since that would be more simply written as $3b_{k-1}/2$ I wonder if you did not intend $b_k= \frac{b_{k-1}}{2+ b_{k-1}}$.

Assuming what you wrote was correct, since $b_0= 1$, $b_1= 1/2+ 1= 3/2$, $b_2= (3/2)/2+ (3/2)= 3/4+ 3/2=3/4+ 6/4=10/4=5/2$... That is not what you write so apparently that is not what you intend.

With $b_k= \frac{b_{k-1}}{2+b_{k-1}}$ and $b_0=1$, $b_1= \frac{1}{2+ 1}= \frac{2}{3}$, $b_2= \frac{\frac{2}{3}}{2+ \frac{2}{3}}= \frac{\frac{2}{3}}{\frac{6}{3}+ \frac{2}{3}}= \frac{\frac{2}{3}}{\frac{8}{3}}= \frac{2}{3}\frac{3}{8}= \frac{2}{8}= \frac{1}{4}$.

That is also not what you have so, no, you are not correct.

I can't respond to the rest of you questions because I do not know whether you intend $b_k= \frac{b_{k-1}}{2}+ b_k= \left(\frac{3}{2}\right)b_{k-1}$ or $b_k= \frac{b_{k-1}}{2+ b_{k-1}}$.

$b_k = \dfrac{b_{k-1}}{2+b_{k-1}}$

$\displaystyle \bigg\{ 1, \dfrac{1}{3}, \dfrac{1}{7}, \dfrac{1}{15}, … , \dfrac{1}{2^{n+1}-1} , … \bigg \}$

## 1. What is meant by the "sequence of bk with explicit formula"?

The sequence of bk with explicit formula refers to a sequence of numbers that can be written in the form of an explicit formula, where each term in the sequence is defined by a specific mathematical expression.

## 2. How is the explicit formula for a sequence of bk determined?

The explicit formula for a sequence of bk can be determined by analyzing the pattern of the sequence and using mathematical techniques such as algebra and calculus to find a formula that can generate each term in the sequence.

## 3. What is the purpose of proving the sequence of bk by mathematical induction?

The purpose of proving the sequence of bk by mathematical induction is to demonstrate that the explicit formula accurately generates each term in the sequence, and to show that the formula holds true for all values of k in the sequence.

## 4. Can the sequence of bk be proven by methods other than mathematical induction?

Yes, the sequence of bk can also be proven by other methods such as direct proof, proof by contradiction, or proof by exhaustion. However, mathematical induction is often the most efficient and widely used method for proving sequences.

## 5. What are the steps involved in proving the sequence of bk by mathematical induction?

The steps involved in proving the sequence of bk by mathematical induction include: stating the base case, assuming the formula holds true for a specific value of k, proving that the formula also holds true for the next value of k, and concluding that the formula holds true for all values of k in the sequence.

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