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 Post subject: Bionomial Expansions help :)Posted: Mon, 6 Feb 2012 02:28:56 UTC
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Joined: Mon, 6 Feb 2012 02:21:29 UTC
Posts: 1
I am really stuck on this and I've spent hours trying to figure out these question sets.

A) Three consecutive coefficients in the expansion of (1+x)^n are in the ratio 6:14:21. Find the value of n.

B) Find the independent term in the [2x + 1 - 1/(2x^2)]^6 (independent term is x^0)

C) In the expansion of (1 + ax)^n the first term is 1, the second term is 24x, and the third term is 252x^2. Find the values of a and n

D) In the expansion of (x + a)^3(x - b)^6, the coefficient of x^7 is -9 and there is no x^8 term. Find a and b.

So I have worked on all four of them:
A) I realized that the coefficients are n C r, n C (r+1), and n C (r+2), and that they are in ratio of 6:14:21. However I am not sure how to find n after this step.

B) I tried substituting a variable y { let y = 2x - 1/(2x^2) } to form (y + 1)^6. I could expand it all out and test each term using the general term of binomial expansion, but that would be very tedious. I am wondering if there is a better solution

C) I simplified C) to (a)(n C 1) = 24 and (a^2)(n C 2) = 252. Now I'm stuck on what to do next

D) I've simplified it to 5b^2 -6ab + a^2 = -3 and -2b^2 +a = 0. I'm not sure how to proceed from this.

Again, any help is much appreciated.

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 Post subject: Re: Bionomial Expansions help :)Posted: Mon, 6 Feb 2012 04:24:36 UTC
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Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12170
Location: Austin, TX
n3rdwannab3 wrote:
I am really stuck on this and I've spent hours trying to figure out these question sets.

A) Three consecutive coefficients in the expansion of (1+x)^n are in the ratio 6:14:21. Find the value of n.

B) Find the independent term in the [2x + 1 - 1/(2x^2)]^6 (independent term is x^0)

C) In the expansion of (1 + ax)^n the first term is 1, the second term is 24x, and the third term is 252x^2. Find the values of a and n

D) In the expansion of (x + a)^3(x - b)^6, the coefficient of x^7 is -9 and there is no x^8 term. Find a and b.

So I have worked on all four of them:
A) I realized that the coefficients are n C r, n C (r+1), and n C (r+2), and that they are in ratio of 6:14:21. However I am not sure how to find n after this step.

B) I tried substituting a variable y { let y = 2x - 1/(2x^2) } to form (y + 1)^6. I could expand it all out and test each term using the general term of binomial expansion, but that would be very tedious. I am wondering if there is a better solution

C) I simplified C) to (a)(n C 1) = 24 and (a^2)(n C 2) = 252. Now I'm stuck on what to do next

D) I've simplified it to 5b^2 -6ab + a^2 = -3 and -2b^2 +a = 0. I'm not sure how to proceed from this.

Again, any help is much appreciated.

You know that , write this out, cancel some terms, then write out the other ratio information and go from there.

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 Post subject: Re: Bionomial Expansions help :)Posted: Mon, 6 Feb 2012 14:14:54 UTC
 Member of the 'S.O.S. Math' Hall of Fame

Joined: Mon, 19 May 2003 19:55:19 UTC
Posts: 7961
Location: Lexington, MA
Hello, n3rdwannab3!

Quote:
(C) In the expansion of , the first term is 1,
the second term is , and the third term is
Find the values of and

I simplified to: .
Now I'm stuck on what to do next.

Hmmm, the next few steps should be obvious . . .

.[1]

.[2]

Substitute [1] into [2] and solve for . . . then solve for

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 Post subject: Re: Bionomial Expansions help :)Posted: Mon, 6 Feb 2012 14:53:41 UTC
 Member of the 'S.O.S. Math' Hall of Fame

Joined: Mon, 19 May 2003 19:55:19 UTC
Posts: 7961
Location: Lexington, MA
Hello again, n3rdwannab3!

Quote:
(B) Find the constant term in .

I tried substituting a variable
Let .to form

I could expand it all out and test each term using the general term of binomial expansion,
. . but that would be very tedious. .I am wondering if there is a better solution.

My method is somewhat less tedious . . . but still a long procedure.

In (a), the constant term is

In (b), we have: .
The constant term is: .

In (c), we have: .[/c
olor]
The constant term is:[color=beige] .

In (d), we have: .
There is no constant term.
The same is true for the subsequent terms.

Therefore, the constant term is: .

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