FP1 Induction X1

I really can't get my head around the harder kind of these summation problems. Especially this factorial one.



I don't think it's likely something like this will appear in the 2016 FP1 paper but I have to make sure I would know how to deal with this just in case.

Thanks




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@seanfm

I have currently shown the summation formula is true for n=1

(1)1=(1+1)!-1
1=(2x1)-1=1

I've assumed true for n=k but not much further than that apart from substituting n=k+1


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What does your substitution look like for n=k+1, i.e what have you done?

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It feels painfully obvious what I have to do next but I can't piece it together


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I have added a new picture, using the student room app atm and it's very buggy.


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If you move the -1 to the very right hand side of the equation at the bottom, can you see anything that you can do to the first two terms?

i tried that and factorised to get

(k+1)![1+(k+1)]-1 = (k+1)!(k+2)-1

If everything else fails take a look at this link
http://madasmaths.com/archive/maths_booklets/further_topics/various/proof_by_induction.pdf

Awesome! You are almost there, there are just... two steps for the proving it for n=k+1 and then the usual 'by mathematical induction'.. stuff.

Wow what an excellent set of resources, I may have to try out some of those papers on the site. Thanks for sharing that with me

I can't see it, its so frustrating, any hints?

Oh, sorry, you are one step away as I did not see the bit after = which was.

[(k+1)!(k+2)] -1.

Remember that you are looking for it to be of the form (n+1)! - 1, where n = k+1..

What can (k+1)! * (k+2) be turned into? (Hint: it's another factorial).

im guessing it gets turned into (k+2)! but I don't really understand why

Precisely.

The definition of (k+2)! for some k is (k+2)*(k+1)*(k)*(k-1)....2*1. and the defintion of (k+1)! is (k+1)*(k)*(k-1)...2*1

so (k+2)*(k+1)! is..

that took a lot of staring to understand, but it is blindingly obvious now.

(k+2)*(k+1)(k)(k-1) is clearly (k+2)!

sorry for the late reply and thank you very much for your help