# Binomial expansion- three non-zero terms find b

I have been doing expansions for what feels like an eternity now. Can someone clarify I’m on the correct path so far with my working. Thanks

Question 9.

My work.

So, next would be to expand the brackets and then compare the co-efficient from the given solution in the question to find b?
(edited 1 year ago)
Original post by KingRich
I have been doing expansions for what feels like an eternity now. Can someone clarify I’m on the correct path so far with my working. Thanks

Question 9.

My work.

So, next would be to expand the brackets and then compare the co-efficient from the given solution in the question to find b?

You have the unknowns n and a as well. Id use the linear and quadratic terms to get them and the cubic term should give you b. Note youll have to expand the (1+ax)^n up to the cubic term.
(edited 1 year ago)
It states up to the first three non-zero terms. Up to the quadratic term should suffice? When you expand with (1-x) the cubic term is found (-x)(x²).

I’m approaching it from the point as there’s no x term in the answer given.
I can equate na-1=0 to find the value of a in terms of n.
Original post by KingRich
I have been doing expansions for what feels like an eternity now. Can someone clarify I’m on the correct path so far with my working. Thanks

Question 9.

My work.

So, next would be to expand the brackets and then compare the co-efficient from the given solution in the question to find b?

I can help you out . My As level teachers just concluded the topic
Original post by KingRich
It states up to the first three non-zero terms. Up to the quadratic term should suffice? When you expand with (1-x) the cubic term is found (-x)(x²).

I’m approaching it from the point as there’s no x term in the answer given.
I can equate na-1=0 to find the value of a in terms of n.

The linear term is
0x
so that gives you the
na-1=0
Then use that with the quadratic coefficient to get n and a. Then use the cubic term to get b.

Youre equating cubic terms to get b. You need to expand (1+ax)^n up to the cubic term as youre multiplying it by (1-x). The
cubic term * 1 - quadratic term * x
gives b.
Original post by Dammie(grateful)
I can help you out . My As level teachers just concluded the topic

This is year 2 work but the expansion taught in As level should work the same way. If you’d like to attempt to answer, then I encourage you to. Obviously don’t give me the answer if you find it.
Original post by mqb2766
The linear term is
0x
so that gives you the
na-1=0
Then use that with the quadratic coefficient to get n and a. Then use the cubic term to get b.

Youre equating cubic terms to get b. You need to expand (1+ax)^n up to the cubic term as youre multiplying it by (1-x). The
cubic term * 1 - quadratic term * x
gives b.

As I thought then equating to 0 to find x.

Mmm, it could be why I couldn’t find the complete answer. I attempted by my approach expanding only to the quadratic term but I found b=3/8

I found a=-(1/4) and n=-4. Although, my working may have been wrong the first time around.

I shall expand to cubic term after eating and see how I go from there
Youre a bit off with those values for a and n, but not hugely. Have a check of what youve done and upload if necessary.
Original post by KingRich
As I thought then equating to 0 to find x.

Mmm, it could be why I couldn’t find the complete answer. I attempted by my approach expanding only to the quadratic term but I found b=3/8

I found a=-(1/4) and n=-4. Although, my working may have been wrong the first time around.

I shall expand to cubic term after eating and see how I go from there
Original post by mqb2766
Youre a bit off with those values for a and n, but not hugely. Have a check of what youve done and upload if necessary.

I kind of got annoyed and tore up and threw in the bin lol

The expansion of (-x) with the quadratic term is really throwing me off with all the brackets and changing directions. ☹️

So far I have…

If you understand my working out. I have used brackets to separate the x-terms for tidying up purposes
Original post by KingRich
I kind of got annoyed and tore up and threw in the bin lol

The expansion of (-x) with the quadratic term is really throwing me off with all the brackets and changing directions. ☹️

So far I have…

If you understand my working out. I have used brackets to separate the x-terms for tidying up purposes

So you have for the linear coefficient
1 - na = 0
so
na = 1
so theyre inversely related. Then for the quadratic coefficientyou have
-na + a^2n(n-1)/2 = 1
So sub for na in both terms and solve for n.
(edited 1 year ago)
Original post by mqb2766
So you have for the linear coefficient
1 - na = 0
so
na = 1
so theyre inversely related. Then for the quadratic coefficientyou have
-na + a^2n(n-1)/2 = 1
So sub for na in both terms and solve for n.

Mmm, I probably did it the most complicated way. I found n=-1/3 and a=-3
Original post by KingRich
Mmm, I probably did it the most complicated way. I found n=-1/3 and a=-3

Looks good. To solve the quadratic part, you simply note
a^2 n = 1/n
which you must have done.
Original post by mqb2766
Looks good. To solve the quadratic part, you simply note
a^2 n = 1/n
which you must have done.

I still haven’t solved the quadratic part.

I keep second guessing myself

I want to say :

Edit:
I must be wrong because I keep getting b=-2 now
(edited 1 year ago)
Original post by KingRich
I still haven’t solved the quadratic part.

I keep second guessing myself

I want to say :

Thought you had solved the quadratic coefficient part to get the previous (correct) values of a and n:
-an + a^2n(n-1)/2 = 1
As
an = 1
a^2n^2 = 1
so
a^2 n = 1/n
then simply sub into the equation and get n
(edited 1 year ago)
Original post by mqb2766
Thought you had solved the quadratic coefficient part to get the previous (correct) values of a and n:
-an + a^2n(n-1)/2 = 1
As
an = 1
a^2n^2 = 1
so
a^2 n = 1/n
then simply sub into the equation and get n

I think I’ll have to come back to this in the morning with a fresh mind. It’s reached it’s limit, not that it has a high limit lol