Science:Math Exam Resources/Courses/MATH220/April 2011/Question 07 (c)

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MATH220 April 2011

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Question 07 (c)
Let $\{b_{n}\}$ be a sequence defined by $b_{1}=2\quad {\text{ and }}\quad b_{n+1}={\frac {b_{n}+{\sqrt {b_{n}}}}{2}}\quad {\text{ for each }}n\in \mathbb {N} .$ Prove that $b_{n+1}\leq b_{n}\quad {\text{ for each }}n\in \mathbb {N} .$

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Use induction!

Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

We will prove this statement by induction on $n$. First, for $n=1$. By definition of the sequence we have that $\displaystyle {\begin{aligned}b_{2}&={\frac {b_{1}+{\sqrt {b_{1}}}}{2}}\\&={\frac {2+{\sqrt {2}}}{2}}\\&\leq {\frac {2+2}{2}}\\&=2\\&\leq b_{1}\end{aligned}}$ which proves the first step of our induction. Let us now assume that the statement is true for all values of $n$ up to $m$ and let us show that the statement holds for $n=m+1$. In part (b) we showed that $\displaystyle b_{n}\geq 1\quad {\text{for all }}n\in \mathbb {N}$ hence we can guarantee that $\displaystyle {\sqrt {b_{n}}}\leq b_{n}\quad {\text{for all }}n\in \mathbb {N}$ and so, we have that $\displaystyle {\begin{aligned}b_{m+1}&={\frac {b_{m}+{\sqrt {b_{m}}}}{2}}\\&\leq {\frac {b_{m}+b_{m}}{2}}\\&=b_{m}\end{aligned}}$ which concludes our proof.

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