Binomial expansion when index is unknown (C2)

How do you do the C2 questions on the binomial expansion where you are given something like (1-3x)^p and given a coefficient of one of the terms?

For example, when (1-3x)^p is expanded the coefficient of x^2 is 54. Given that p>0, use this information to find:
a) the value of the constant p
b) the coefficient of x

I can do normal binomial expansion questions and questions where you have for example (1-kx)^4 or something, but how would you go about doing this type of question?

Reply 1

Glutamic Acid

Use p when calculating the binomial coefficients:

$(1-3x)^p = 1 - \dbinom{p}{1}(3x) + \dbinom{p}{2}(3x)^2...$

Where $\dbinom{p}{1} = p$
and $\dbinom{p}{2} = \dfrac{p(p-1)}{2}$

Reply 2

qgujxj39

Well the coefficient is going to be 9(pC2). You then find the value of p such that that coefficient equals 4.

Reply 3

Muppety_Kid

Glutamic Acid

Use p when calculating the binomial coefficients:

$(1-3x)^p = 1 - \dbinom{p}{1}(3x) + \dbinom{p}{2}(3x)^2...$

Where $\dbinom{p}{1} = p$
and $\dbinom{p}{2} = \dfrac{p(p-1)}{2}$

Shouldn't it be over 1!, 2!, 3!... though?

Maybe I've done it differently. I always used the formula:

$(1-3x)^p=1+p(-3x)+\frac{p(p-1)(-3x)^2}{2!}+\frac{p(p-1)(p-2)(-3x)^3}{3!}+...$

Hope this helps :wink: