$$ \left(I^{ ext {st }} ight. $$ Session 2022) If the ratio between the fourth term from the beginning to the fourth term from the end in the expansion of $$ \left(\frac{a}{x}+\frac{x}{a} ight)^{2 n} $$ equals $$ a^{4}: x^{4} $$ , then $$ n=\cdots \cdots $$ $$ \begin{array}{llll} ext { (a) } 2 & ext { (b) } 4 & \left. ext { (where } x \in \mathbb{R}^{*}, a \in \mathbb{R}^{*} ight)\end{array} $$

Question

$$ \left(I^{	ext {st }}ight. $$ Session 2022) If the ratio between the fourth term from the beginning to the fourth term from the end in the expansion of $$ \left(\frac{a}{x}+\frac{x}{a}ight)^{2 n} $$ equals $$ a^{4}: x^{4} $$ , then $$ n=\cdots \cdots $$ $$ \begin{array}{llll}	ext { (a) } 2 & 	ext { (b) } 4 & \left.	ext { (where } x \in \mathbb{R}^{*}, a \in \mathbb{R}^{*}ight)\end{array} $$

UpStudy AI · Accepted Answer

To solve the problem, consider the expansion of $$\left(\frac{a}{x} + \frac{x}{a}\right)^{2n}$$. We need to find the value of $$n$$ such that the ratio of the fourth term from the beginning to the fourth term from the end is $$a^4 : x^4$$.

### Step 1: General Term in the Expansion
The general term in the expansion of $$\left(\frac{a}{x} + \frac{x}{a}\right)^{2n}$$ is given by:

$$
T_r = \binom{2n}{r} \left(\frac{a}{x}\right)^r \left(\frac{x}{a}\right)^{2n - r}
$$

Simplifying:

$$
T_r = \binom{2n}{r} a^{2r - 2n} x^{2n - 2r}
$$

### Step 2: Identifying the Relevant Terms
- **Fourth Term from the Beginning:** This corresponds to $$r = 3$$.

$$
T_3 = \binom{2n}{3} a^{6 - 2n} x^{2n - 6}
$$

- **Fourth Term from the End:** Since there are $$2n + 1$$ terms in total, this corresponds to $$r = 2n - 2$$.

$$
T_{2n-2} = \binom{2n}{2n-2} a^{2n - 4} x^{-2n + 4}
$$

### Step 3: Setting Up the Ratio
The ratio of the fourth term from the beginning to the fourth term from the end should be equal to $$\frac{a^4}{x^4}$$:

$$
\frac{T_3}{T_{2n-2}} = \frac{\binom{2n}{3} a^{6 - 2n} x^{2n - 6}}{\binom{2n}{2} a^{2n - 4} x^{-2n + 4}} = \frac{\binom{2n}{3}}{\binom{2n}{2}} \cdot a^{10 - 4n} x^{4n - 10} = \frac{a^4}{x^4}
$$

### Step 4: Solving for $$n$$
Equating the exponents of $$a$$ and $$x$$:

1. $$10 - 4n = 4$$ $$\Rightarrow$$ $$n = \frac{6}{4} = 1.5$$
2. $$4n - 10 = -4$$ $$\Rightarrow$$ $$n = \frac{6}{4} = 1.5$$

However, $$n$$ must be an integer. Testing $$n = 2$$:

- For $$n = 2$$, the ratio simplifies to:

$$
\frac{4a^2/x^2}{4x^2/a^2} = \frac{a^4}{x^4}
$$

This satisfies the given condition.

### Conclusion
The correct value of $$n$$ is **2**.

**Answer:**  
**(a) 2**
$$ n = 2 $$

Session 2022) If the ratio between the fourth term from the beginning to the
fourth term from the end in the expansion of equals
, then

Upstudy AI Solution

Answer

Solution

Step 1: General Term in the Expansion

Step 2: Identifying the Relevant Terms

Step 3: Setting Up the Ratio

Step 4: Solving for

Conclusion

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Session 2022) If the ratio between the fourth term from the beginning to the fourth term from the end in the expansion of equals , then