Solving for a in Binomial Expansion: Find Possible Values

[thornluke]

The coefficient of x in the expansion of [x+(1/ax^2)]^7 is 7/3. Find the possible values of a.

1. Rewrite (x + 1/(ax^2))^7 = x^(-14) (x^3 + 1/a)^7.
So, we need to find the coefficient of x^15 from (x^3 + 1/a)^7.

2. Using the Binomial Theorem, we have
(x^3 + 1/a)^7 = Σ(k = 0 to 7) C(7, k) (x^3)^(7 - k) (1/a)^k.
......= Σ(k = 0 to 7) C(7, k) x^(21 - 3k) (1/a)^k.

3. So, we need 21 - 3k = 15 ==> k = 2.

Thus, we have (1/a)^2 = 7/3
==> a = ±√(3/7).

The problem is, I do not understand the steps. Help please?

 

[grzz]

Rewrite (x + 1/(ax^2))^7 = x^(-14) (x^3 + 1/a)^7.

First \frac{1}{x^{2}} is taken out as a common factor. Note that this factor is to the power of 7.

 

[thornluke]

First \frac{1}{x^{2}} is taken out as a common factor. Note that this factor is to the power of 7.

Why do we need to find the coefficient of x^15 from (x^3 + 1/a)^7?

 

[Ray Vickson]

Why do we need to find the coefficient of x^15 from (x^3 + 1/a)^7?

How else could you get x^1 when multiplying x^(-14) and x^n?

RGV

 

[thornluke]

How else could you get x^1 when multiplying x^(-14) and x^n?

RGV

Oh right.. Thank you! :)

 

[HallsofIvy]

I wouldn't have factored out the x^{14}. Each term of (u+ v)^n is of the form
\begin{pmatrix}n \\ i\end{pmatrix}u^iv^{n-i}

Here, u= x and a= 1/ax^2= (1/a)x^{-2} so that formula would be
\begin{pmatrix}7 \\ i\end{pmatrix}a^{i- 7}(x^i)(x^{2i- 14})
and we want the power of x, i+ 2i- 14= 1. That gives 3i= 15 or i= 5.

It looks to me like you have forgotten the binomial coefficient. I get an integer value for a.

 

[thornluke]

I wouldn't have factored out the x^{14}. Each term of (u+ v)^n is of the form
\begin{pmatrix}n \\ i\end{pmatrix}u^iv^{n-i}

Here, u= x and a= 1/ax^2= (1/a)x^{-2} so that formula would be
\begin{pmatrix}7 \\ i\end{pmatrix}a^{i- 7}(x^i)(x^{2i- 14})
and we want the power of x, i+ 2i- 14= 1. That gives 3i= 15 or i= 5.

It looks to me like you have forgotten the binomial coefficient. I get an integer value for a.

I don't understand how \begin{pmatrix}7 \\ i\end{pmatrix}a^{i- 7} can come from u^iv^{n-i}

If it's n-i shouldn't it be a^7-i and not a^i-7?