Algebra help for proof by induction
I’m doing part c
Here is the answer in the book
I’m ok until the penultimate line. Where does that extra bit of addition come from? the ((k+1](3k+8))/((k+2)(k+3))
Here is the answer in the book
I’m ok until the penultimate line. Where does that extra bit of addition come from? the ((k+1](3k+8))/((k+2)(k+3))
Looks like an error. The "+" in that line should be an "=" sign.
Now as to why the equality holds, it would be a factorizing cubic exercise for you (as maths people like to say/dread to see).
Now as to why the equality holds, it would be a factorizing cubic exercise for you (as maths people like to say/dread to see).
Original post by tonyiptony
Looks like an error. The "+" in that line should be an "=" sign.
Now as to why the equality holds, it would be a factorizing cubic exercise for you (as maths people like to say/dread to see).
Now as to why the equality holds, it would be a factorizing cubic exercise for you (as maths people like to say/dread to see).
Yes, I suspected as much as that extra bit would have had to = 0. Ok - now to factorising a cubic.
Yes, it factorises.
Original post by maggiehodgson
Yes, it factorises.
Couple of things to have in your pocket for a similar question
* Doing induction on the sum to n-1 or n-2 and adding on the next term would probably simplify the algebra a bit. Not worked it through but maybe try it?
* You could do differences/telescoping here as the partial fractions if of the "right form". Again sum to n-1 as its probably a bit simpler algebra. The question probably asked for induction but if it was an "or otherwise ..." type question its good to consider as a canddate.
(edited 8 months ago)
Original post by maggiehodgson
Yes, it factorises.
Note that this is a "if it works, then k+1 must be a factor" scenario so you effectively know one factor before you start. It's a bit of a 'cheat' (because you couldn't do it if you didn't know the rough form of where you're hoping to end up), but you should take advantage where you can.
Original post by mqb2766
Couple of things to have in your pocket for a similar question
* Doing induction on the sum to n-1 or n-2 and adding on the next term would probably simplify the algebra a bit. Not worked it through but maybe try it?
* You could do differences/telescoping here as the partial fractions if of the "right form". Again sum to n-1 as its probably a bit simpler algebra. The question probably asked for induction but if it was an "or otherwise ..." type question its good to consider as a canddate.
* Doing induction on the sum to n-1 or n-2 and adding on the next term would probably simplify the algebra a bit. Not worked it through but maybe try it?
* You could do differences/telescoping here as the partial fractions if of the "right form". Again sum to n-1 as its probably a bit simpler algebra. The question probably asked for induction but if it was an "or otherwise ..." type question its good to consider as a canddate.
I'd probably have rewritten the RHS as 3 - (Ax+B)/((k+1)(k+2)); the next simplification would be to fully partial fraction it - interesting to note that this recaptures the telescoping solution.
Original post by mqb2766
Couple of things to have in your pocket for a similar question
* Doing induction on the sum to n-1 or n-2 and adding on the next term would probably simplify the algebra a bit. Not worked it through but maybe try it?
* You could do differences/telescoping here as the partial fractions if of the "right form". Again sum to n-1 as its probably a bit simpler algebra. The question probably asked for induction but if it was an "or otherwise ..." type question its good to consider as a canddate.
* Doing induction on the sum to n-1 or n-2 and adding on the next term would probably simplify the algebra a bit. Not worked it through but maybe try it?
* You could do differences/telescoping here as the partial fractions if of the "right form". Again sum to n-1 as its probably a bit simpler algebra. The question probably asked for induction but if it was an "or otherwise ..." type question its good to consider as a canddate.
Thanks. It was a question on proof by induction.
Original post by DFranklin
Note that this is a "if it works, then k+1 must be a factor" scenario so you effectively know one factor before you start. It's a bit of a 'cheat' (because you couldn't do it if you didn't know the rough form of where you're hoping to end up), but you should take advantage where you can.
Yes. That's exactly what I did. I knew the (k+1)s had to cancel so I did algebra division to be left with a quadratic.
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