The Student Room Group
Uni students - Complete our survey >>
Uni students - Complete our survey >>
Uni students - Complete our survey >>

Binomial Proof

So I have this question asking for a proof of a relationship between the second and third co-efficients of the Binomial Expansion:

(a+x)n(a+x)^n

are always equal to:

(r+1)an−r\frac{(r + 1)a}{n - r}

The coefficients follow nn and n(n+1)2\frac{n(n + 1)}{2}

But how can I pull these together to one simple proof?

Completely stuck - just a hint required, am I on the right tracks with looking at series for the Pascal's Triangle sequences?

:frown:
Binomial Proof question.jpg45.6KB
Just a point: They aren't asking to prove a relationship between the second and third coefficients. They're asking to prove it between the rthr^{th} and (r+1)th(r+1)^{th}.
Reply 2
Avatar for Gmart
Gmart
OP
13
Does that mean induction?
Original post by Gmart
Does that mean induction?


that looks promising.
Original post by Gmart
Does that mean induction?


I don't think you need to go that far. You can use the formula for the binomial expansion, I think it has to do with the formula for (nr)n \choose r.
(edited 8 years ago)
Reply 5
Avatar for Zacken
Zacken
22
Original post by EricPiphany
I don't think you need to go that far. You can use the formula for the binomial expansion, I think it has to do with the formula for (nr)n \choose r.


Original post by Gmart
Does that mean induction?


I concur with this, just use the factorial definition of (nr)=n!r!(n−r)!\displaystyle {n \choose r} = \frac{n!}{r!(n-r)!} (I may have written that down incorrectly, so just check to be sure)
Reply 6
Avatar for Gmart
Gmart
OP
13
So the a vale takes care of itself, but the fraction I have down to:

r!(n−r)!(n−1)!\frac{r!(n-r)!}{(n-1)!}

But what is my next step? I need:

r+1n−1\frac{r + 1}{n - 1}

:confused:
Reply 7
Avatar for Zacken
Zacken
22
Original post by Gmart
So the a vale takes care of itself, but the fraction I have down to:

r!(n−r)!(n−1)!\frac{r!(n-r)!}{(n-1)!}

But what is my next step? I need:

r+1n−1\frac{r + 1}{n - 1}

:confused:


Coefficient of rth term:

α=an−r⋅n!r!(n−r)!\displaystyle \alpha = a^{n-r} \cdot \frac{n!}{r!(n-r)!}

Coefficient of (r+1)th term:

β=an−r−1⋅n!(r+1)!(n−r−1)!\displaystyle \beta = a^{n-r - 1} \cdot \frac{n!}{(r+1)!(n-r-1)!}

So, what can you say about αβ\dfrac{\alpha}{\beta}?
Reply 8
Avatar for Gmart
Gmart
OP
13
Thank you so much :smile:
Reply 9
Avatar for Zacken
Zacken
22
Original post by Gmart
Thank you so much :smile:


Very welcome.
Original post by Zacken
Very welcome.


Which module is this from?


Posted from TSR Mobile
Reply 11
Avatar for Zacken
Zacken
22
Original post by anoymous1111
Which module is this from?


Posted from TSR Mobile


This is IB - but it's C1/2 knowledge.
Original post by Zacken
Coefficient of rth term:

α=an−r⋅n!r!(n−r)!\displaystyle \alpha = a^{n-r} \cdot \frac{n!}{r!(n-r)!}

Coefficient of (r+1)th term:

β=an−r−1⋅n!(r+1)!(n−r−1)!\displaystyle \beta = a^{n-r - 1} \cdot \frac{n!}{(r+1)!(n-r-1)!}

So, what can you say about αβ\dfrac{\alpha}{\beta}?


Where do I go from here?


Posted from TSR Mobile
Reply 13
Avatar for Zacken
Zacken
22
Original post by anoymous1111
Where do I go from here?


Posted from TSR Mobile


Simplify the quotient of factorials.

αβ=an−ran−r−1⋅n!r!(n−r)!⋅(r+1)!(n−r−1)!n!=⋯\displaystyle \frac{\alpha}{\beta} = \displaystyle \frac{a^{n-r}}{a^{n-r-1}} \cdot \frac{n!}{r!(n-r)!} \cdot \frac{(r+1)!(n-r-1)!}{n!} = \cdots

Quick Reply

Related discussions

Latest

Trending

Trending

Articles for you