Science:Math Exam Resources/Courses/MATH220/December 2011/Question 06 (b)

MATH220 December 2011

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Question 06 (b)
Use mathematical induction to prove the following statement: For all $n\in \mathbb {N}$, $12^{2n-1}+11^{n+1}$ is a multiple of $133$.

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
This is a routine induction problem. You might want to use the fact that $12^{2}=144=133+11$

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. First, we prove the claim true when $n=1$. In this case, $\displaystyle 12^{2n-1}+11^{n+1}=12^{2(1)-1}+11^{1+1}=12+121=133$ which is clearly divisible by 133. We assume the claim is true for $n=k$. For $n=k+1$, we have ${\begin{aligned}12^{2(k+1)-1}+11^{(k+1)+1}&=12^{2k+1}+11^{k+2}\\&=12^{2}\cdot 12^{2k-1}+11\cdot 11^{k+1}\\&=144\cdot 12^{2k-1}+11\cdot 11^{k+1}\\&=(133+11)12^{2k-1}+11\cdot 11^{k+1}\\&=133\cdot 12^{2k-1}+11\cdot 12^{2k-1}+11\cdot 11^{k+1}\\&=133\cdot 12^{2k-1}+11(12^{2k-1}+11^{k+1})\end{aligned}}$ Now, in the last line, the first summand is divisible by 133 and the second summand is also divisible by 133 by the induction hypothesis. Hence we must have that $12^{2(k+1)-1}+11^{(k+1)+1}$ is divisible by 133 as required.

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