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# Induction, is this correct? watch

1. So I have to prove this
for

so the base case

then inductive hypothesis n=k

to show that it implies n=k+1

=>

=>

by since .
So by induction, for .

Is this correct?
2. Looks good to me.
3. I'd advise being a bit clearer when presenting. Your statements are written like this: a = b > c = d, which is confusing to say the least, if not plain wrong.
4. (k!)(k+1)>2^{k}(k+1)
this seems dodgy to me but what do i know
(k!)(k+1)>2^{k+1} ?
5. (Original post by spex)
I'd advise being a bit clearer when presenting. Your statements are written like this: a = b > c = d, which is confusing to say the least, if not plain wrong.
What's wrong with it? It might look a bit ugly, but I don't think it's wrong.
6. (Original post by Intergrate this u F##*...)
(k!)(k+1)>2^{k}(k+1)
this seems dodgy to me but what do i know
(k!)(k+1)>2^{k+1} ?
7. (Original post by generalebriety)
What's wrong with it? It might look a bit ugly, but I don't think it's wrong.
Well you should be able to to read a statement on either side of an equals sign and it make sense.

The second line suggests that 4! =... = 16.
8. (Original post by spex)
Well you should be able to to read a statement on either side of an equals sign and it make sense.

The second line suggests that 4! =... = 16.
Well, yes, if it's a chain of equality, but the "<" sign breaks that chain. If you read it from left to right it says "4! equals 24, which is less than 2^4, which equals 16".
9. Thanks. Its just in some cases its hard to know how much to write.

A good example is Bernoulli's inequality.

for all non-negative integer values of n and real number x>-1.

So base step, therefore n=0.

which is trivially true.

Inductive step let n=k.

so to show that this implies n=k+1

(no sign change since ((1+x)>0)
=>

Since ( and ) =>.

So by induction, proven.

But I don't know about that last bit. Its sort of obvious, but would I need to go into more detail.
10. That's fine. (It's not really that obvious.)
11. I don't understand what comes after the = sign on this line:

12. (Original post by mcp2)
I don't understand what comes after the = sign on this line:

Well, kx + x = (k+1)x. Look at the number of "x" on the left hand side; you have k of them, plus one of them. That's k+1 of them.
13. Understood that but where did 1+(k+1)x come from after the second > sign?
14. (Original post by mcp2)
Understood that but where did 1+(k+1)x come from after the second > sign?
Recall three fundamental properties of inequalities:
- squares are non-negative
- multiplying both sides by a positive number preserves the inequality
- adding the same number to both sides preserves the inequality

x^2 ≥ 0 => kx^2 ≥ 0 when k ≥ 0 [multiplying by a positive number]
=> kx^2 + (k+1)*x + 1 ≥ 1 + (k+1)*x [adding 1 + (k+1)x...]

We did this because 1 + (k+1)*x was our goal by induction.
15. (Original post by AsakuraMinamiFan)
=> kx^2 + (k+1)*x + 1 ≥ 1 + (k+1)*x [adding 1 + (k+1)x...]

We did this because 1 + (k+1)*x was our goal by induction.
Thanks! That's what was the bit that was troubling me! This should help when I actually do induction!

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