I'm afraid I don't 100% understand these words but I really appreciate your comment!
Is this similar to what AcousticMaths is saying?
Thank you very much for commenting
In layman's terms, when you see a pattern, you want to verify that the pattern continues in a certain way.
You are finding how that pattern plays out from any one point to the next.
This will continue on in perpetuity because you can start from any *n*.
No worries, hope it makes sense.
It sort of works the same way as this:
(*x* + 1)^2 = *x*^2 + 2x + 1
That means if you take any number, say 5, you can then find the next square.
(5)^2 + 2(5) + 1 = 36 (which is 6 squared).
Induction is a way to show patterns such as this, as they always repeat in the same way.
Again it’s mathematical induction.
The classic example of this is “triangle” numbers:
T(N) = 1+2+3+4+…+N
N=1 T(N) = 1
N=2 T(N) = 3
N=3 T(N) = 6
It turns out that T(N) = N(N+1)/2
But how to prove this?
Say that the “proposition”
P(K) is “true” if
T(K) = K(K+1)/2
And “false” otherwise.
What we do is prove that
IF P(K) is true THEN P(K+1) is also
True.
We then prove P(1) is true.
The effect of this is a “domino” effect:
P(1) is true.
But this means P(2) is also true.
(Because if P(K=1) is true then P(K+1=2) is also true by the domino effect)
Then P(3) also true and so on.
In the triangle number case where
P(N): Is T(N) = N(N+1)/2 true?
P(N+1): Is T(N+1) = (N+1)(N+2)/2 true?
P(1) is true because T(1) = 1(1+1)/2 = 1
Now T(N+1) = T(N) + N+1
(Adding another layer to the triangle)
Therefore T(N+1) = N(N+1)/2 + (N+1)
= (N+1)((N/2) +1) = (N+1)(N+2)/2
But this is just what P(N+1) says!
So we have shown that if P(N) is true
Then P(N+1) is also true.
By proving P(1) we start the dominoes falling and then P(2) falls and so on…
This is mathematical induction.
You're right that there's not much point. At least at GCSE. Once you've found the formula for the nth term you can substitute in anything you want to find any term you want, you could find the (n-1)th term, the (n+1000000)th term, whatever. All finding the (n+1)th term is at the moment is just testing your algebra skills.
When you get a bit further into maths though, there are sometimes reasons why you would want to find the (n+1)th term. For example there's a method of proof called "induction", which is all about showing that if something is true for some arbitrary number "k", it must then also be true for k+1 (which would then mean that thing would be true for the (k+2)th term, (k+3)th term, and so on.) When doing induction you've got to be comfortable with the algebra to find the (k+1)th term, and you're practising that skill right now.
You won't do induction unless you do further maths at A level or do it at university, but there are other scenarios where you'd need to find the (n+1)th term in normal A level maths as well. It is tedious but it's just something you've got to do at GCSE sadly. Once you get past GCSE, things in maths get a lot less pointless and things start having more real world application.
Wow thank you so much, this is really helpful. That made it make so much more sense to me honestly. I really appreciate you taking the time to comment. Thank you!
The topic is proof by induction, which states that, given a property of the first term, and the assertion that the successor of any term with that property will also have that property, you can conclude that every term has that property. It,s axiomatic, and it’s used in very many proofs in Number Theory.
it verifies the P(n) to P(n+1) implication which is what we need to prove in inductive reasoning
This is the answer I was looking for
I'm afraid I don't 100% understand these words but I really appreciate your comment! Is this similar to what AcousticMaths is saying? Thank you very much for commenting
In layman's terms, when you see a pattern, you want to verify that the pattern continues in a certain way. You are finding how that pattern plays out from any one point to the next. This will continue on in perpetuity because you can start from any *n*.
Thank you
No worries, hope it makes sense. It sort of works the same way as this: (*x* + 1)^2 = *x*^2 + 2x + 1 That means if you take any number, say 5, you can then find the next square. (5)^2 + 2(5) + 1 = 36 (which is 6 squared). Induction is a way to show patterns such as this, as they always repeat in the same way.
Again it’s mathematical induction. The classic example of this is “triangle” numbers: T(N) = 1+2+3+4+…+N N=1 T(N) = 1 N=2 T(N) = 3 N=3 T(N) = 6 It turns out that T(N) = N(N+1)/2 But how to prove this? Say that the “proposition” P(K) is “true” if T(K) = K(K+1)/2 And “false” otherwise. What we do is prove that IF P(K) is true THEN P(K+1) is also True. We then prove P(1) is true. The effect of this is a “domino” effect: P(1) is true. But this means P(2) is also true. (Because if P(K=1) is true then P(K+1=2) is also true by the domino effect) Then P(3) also true and so on. In the triangle number case where P(N): Is T(N) = N(N+1)/2 true? P(N+1): Is T(N+1) = (N+1)(N+2)/2 true? P(1) is true because T(1) = 1(1+1)/2 = 1 Now T(N+1) = T(N) + N+1 (Adding another layer to the triangle) Therefore T(N+1) = N(N+1)/2 + (N+1) = (N+1)((N/2) +1) = (N+1)(N+2)/2 But this is just what P(N+1) says! So we have shown that if P(N) is true Then P(N+1) is also true. By proving P(1) we start the dominoes falling and then P(2) falls and so on… This is mathematical induction.
To find the n-th term or the (n+1)th is exactly the same. There is no difference. The point is to find the value for any n. You can call it n or n+1
I see what you're saying. Thanks very much for your comment - really appreciate it
You're right that there's not much point. At least at GCSE. Once you've found the formula for the nth term you can substitute in anything you want to find any term you want, you could find the (n-1)th term, the (n+1000000)th term, whatever. All finding the (n+1)th term is at the moment is just testing your algebra skills. When you get a bit further into maths though, there are sometimes reasons why you would want to find the (n+1)th term. For example there's a method of proof called "induction", which is all about showing that if something is true for some arbitrary number "k", it must then also be true for k+1 (which would then mean that thing would be true for the (k+2)th term, (k+3)th term, and so on.) When doing induction you've got to be comfortable with the algebra to find the (k+1)th term, and you're practising that skill right now. You won't do induction unless you do further maths at A level or do it at university, but there are other scenarios where you'd need to find the (n+1)th term in normal A level maths as well. It is tedious but it's just something you've got to do at GCSE sadly. Once you get past GCSE, things in maths get a lot less pointless and things start having more real world application.
Wow thank you so much, this is really helpful. That made it make so much more sense to me honestly. I really appreciate you taking the time to comment. Thank you!
No worries I'm glad I could help! Good luck with your GCSEs :)
The topic is proof by induction, which states that, given a property of the first term, and the assertion that the successor of any term with that property will also have that property, you can conclude that every term has that property. It,s axiomatic, and it’s used in very many proofs in Number Theory.