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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.
F324D137-F2E4-4AC6-9DE4-DA7993BAC494.jpeg

My work.
0A86425E-2F39-4D9F-A22C-98BD9F202DF2.jpeg

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.
F324D137-F2E4-4AC6-9DE4-DA7993BAC494.jpeg

My work.
0A86425E-2F39-4D9F-A22C-98BD9F202DF2.jpeg

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)
Reply 2
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.
F324D137-F2E4-4AC6-9DE4-DA7993BAC494.jpeg

My work.
0A86425E-2F39-4D9F-A22C-98BD9F202DF2.jpeg

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.
Reply 5
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.
Reply 6
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
Reply 8
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…

70357DA4-9643-459F-A3BE-EEF4BC6DC904.jpeg

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…

70357DA4-9643-459F-A3BE-EEF4BC6DC904.jpeg

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)
Reply 10
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.
Reply 12
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 :
7D4241A3-1490-4BBE-A60A-2FD3DB0E85D7.jpeg

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 :
7D4241A3-1490-4BBE-A60A-2FD3DB0E85D7.jpeg

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)
Reply 14
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

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