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Question: Use the method of mathematical induction to prove that 4^n + 15n -1 is divisible by 9. n is a positive integer. (OK!)

True for n = 1

P(1) ......... = 9A

 

True for n = k

If P(k) = 4k + 15k - 1 = 9A 

 

Then it should be true that for n = (k + 1)

If P(k+1) = 4k+1 + 15k + 15 - 1= 9B

==> 3(4)k + 15 + (4k + 15k - 1)

==> 3(4)k + 15 + 9A

==> 3(4+ 5 + 3A)

 

3 =/= 9

 

6 marks question, why did I derived 3 instead of 9. Someone help? HL Maths IB. I hate it when I follow steps and then end up backed in a corner.

Guest Oct 13, 2017
 #1
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There could be something obvious that I'm not seeing either, but failing that, a second induction proof to show that 4^k + 5 is divisible by 3 works.

Guest Oct 13, 2017
 #2
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P(n) = 4n + 15n - 1.       P(1) = 18.   Divisible by 9

 

P(n+1) = P(n) + Q(n).      (A)         where Q(n) = 3*4n + 15

 

Q(n) = 3*4n + 15.    Q(1) = 27. Divisible by 9

 

Q(n+1) = 3*4n+1 + 15.    Q(n+1) = 12*4n + 15  Q(n+1) = 9*4n + Q(n).

 

Hence if Q(n) divisible by 9 then Q(n+1) divisible by 9 

 

Hence Q(n) divisible by 9 (because Q(1) also divisible by 9)

 

So if P(n) divisible by 9 then P(n+1) divisible by 9 (because both terms on RHS of A are divisible by 9).

 

Since P(1) is also divisible by 9 then P(n) is divisible by 9.  QED.

Alan  Oct 13, 2017
edited by Alan  Oct 13, 2017

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