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[M2] Mathematical Induction

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Mathematical Induction

Please solve the following question.
Thank you for your kind assistance

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Where did you get stuck?

If you just do it, you can get (a) done. Now this is tricky because there's both n and r, but let this be an intro to M2 algebra. Remember you're only showing the proposition is true for n=1 and n=k+1 (assuming n=k is true). Whatever r is is irrelevant.

(b) should be obvious. In fact, if you get stuck at (a) you still can attempt (b).

(c) is interesting. Focus on the last term of the required expression. How do you make (2n-1)3^(n-1) happen with the last terms in the proposition in (a) and the given fact in (c)? This is another intro to M2 algebra - you want to forcibly make things line up nicely. The simplification part is again very algebra-heavy. Remember that 3^(n+1) = 3*3^n. Also sidenote, the given fact in (c) is the formula to sum of geometric progression, which you'll learn in F.6.

Reply here (and preferably send me a PM) if you really get stuck, and want a full solution.

This question is a bit annoying, but you can complete this if you just push through and be careful, which is what you'll need for M2.

[ 本帖最後由 風之男 於 2020-9-13 04:50 AM 編輯 ]

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回覆 2# 風之男 的帖子

Thank you very much

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