Science:Math Exam Resources/Courses/MATH220/December 2009/Question 05 (c)

MATH220 December 2009

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Question 05 (c)
Use mathematical induction to prove the following statements: For all $n\in \mathbb {N}$, $\sum _{j=1}^{n}j^{3}>{\frac {1}{4}}n^{4}.$

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
The proof writes itself after you set it up correctly. Remember to apply mathematical induction on a statement $\displaystyle S(n)$ you must do three things: 1. Prove $\displaystyle S(1)$ is true 2. Assume $\displaystyle S(k)$ is true for some natural number k 3. Show that $\displaystyle S(k+1)$ is true based on the assumption above.

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

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Let $\displaystyle S(n)$ be the statement that $\sum _{j=1}^{n}j^{3}>{\frac {n^{4}}{4}}.$ We prove this statement true for all natural numbers using mathematical induction. Notice that $\displaystyle S(1)$ is true since $\sum _{j=1}^{1}j^{3}=1>{\frac {1}{4}}.$ Now, we assume that $\displaystyle S(k)$ is true for some $k$ and show that $\displaystyle S(k+1)$ is true. When $\displaystyle n=k+1$, we can start with the induction hypothesis: $\sum _{j=1}^{k}j^{3}>{\frac {k^{4}}{4}}.$ To show $\displaystyle S(k+1)$ is true, notice that : ${\begin{aligned}\sum _{j=1}^{k+1}j^{3}&=\sum _{j=1}^{k}j^{3}+(k+1)^{3}\\&>{\frac {k^{4}}{4}}+(k+1)^{3}\\&={\frac {(k+1-1)^{4}+4(k+1)^{3}}{4}}\\&={\frac {(k+1)^{4}-4(k+1)^{3}+6(k+1)^{2}-4(k+1)+1+4(k+1)^{3}}{4}}\\&={\frac {(k+1)^{4}+6(k+1)^{2}-4(k+1)+1}{4}}\\&={\frac {(k+1)^{4}+6k^{2}+12k+6-4k-4+1}{4}}\\&={\frac {(k+1)^{4}+6k^{2}+8k+3}{4}}\\&>{\frac {(k+1)^{4}}{4}}\end{aligned}}$ which shows that $\displaystyle S(k+1)$ is true. Hence $\displaystyle S(n)$ is true for all natural numbers $n$ by the principle of mathematical induction.

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