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# Proof By Induction, FP1 watch

1. could i have some help with edexcel FP1, ex6A question 10. I have started of by doing the n=1 case, and then assume n=k for k=all positive integers...

then i am confused when it comes to substituting in the k+1 for 2n, and also how to finish off the proof, because i can't seem to re-arrange mine to get back to the original K+1 formula
2. It is easier if you post the question
3. Here's the question.

Prove by mathematical induction for n \in Z^+

4. When you replace k with k + 1;
On the left the sum goes up to 2k + 2
which means you need to include two new terms on the right...
(2k + 1)2 and (2k + 2)2

And then you have to show that the rhs including the new terms is equivalent to

1/3( k + 1 )( 2k + 3 )( 4k + 5 )

which you can do by expanding and comparing rather than factorising
5. (Original post by SubAtomic)
Here's the question.

Prove by mathematical induction for n \in Z^+

Use n for the proof
Let the sum be true for n
that is

for n+1

Factorize the terms in the brackets with f.e the quadratic formula

where and is the two roots of the
6. I understand the method, but could you please explain, why I have to add two new terms, (2k + 1)2 and (2k + 2)2
but could you explain where the (2k + 1)^2 term comes from. I thought that the only extra term was (2k + 2)^2.
7. (Original post by Jaisal Patel)
I understand the method, but could you please explain, why I have to add two new terms,

then your sum to k+1, which comes from replacing k with k+1, is

8. thanks,
9. (Original post by ztibor)
Use n for the proof
Let the sum be true for n
that is

for n+1

Factorize the terms in the brackets with f.e the quadratic formula

where and is the two roots of the
I've always being curious to these FP1 questions, what is the method of algebraic manipulation used here Is this instantaneous, or are there more steps, what exactly has happened here, as with every proof by induction question its not the context and instructions that throw me off, its the algebraic manipulation.
10. (Original post by Robbie242)
I've always being curious to these FP1 questions, what is the method of algebraic manipulation used here Is this instantaneous, or are there more steps, what exactly has happened here, as with every proof by induction question its not the context and instructions that throw me off, its the algebraic manipulation.
It is instant, it's just factoring out from both terms.
11. (Original post by Noble.)
It is instant, it's just factoring out from both terms.
And for most induction questions would you use this method of factorisation (i.e the series ones) I can see where the terms come from now though
12. (Original post by Robbie242)
I've always being curious to these FP1 questions, what is the method of algebraic manipulation used here Is this instantaneous, or are there more steps, what exactly has happened here, as with every proof by induction question its not the context and instructions that throw me off, its the algebraic manipulation.
It was a factoring out as Noble. wrote
And arranging after that

the two roots of quadratic equation

that is and
so the term will be

Which is the formula for n+1 and the proof is completed.
13. (Original post by ztibor)
It was a factoring out as Noble. wrote
And arranging after that

the two roots of quadratic equation

that is and
so the term will be

Which is the formula for n+1 and the proof is completed.
Wow thanks so much, I never knew the quadratic step appreciate the alternate method!

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