If the `2^("nd"),3^("rd")" and "4^("th")` terms in the expansion of `(a+x)^(n)` are respectively 240, 720, 1080, find a, x, n.

Answer

To solve the problem, we need to find the values of \( a \), \( x \), and \( n \) given that the second, third, and fourth terms in the expansion of \( (a + x)^n \) are 240, 720, and 1080, respectively.

### Step-by-Step Solution:

1. **Identify the Terms**:
The \( k^{th} \) term in the binomial expansion of \( (a + x)^n \) is given by:
\[ T_k = \binom{n}{k-1} a^{n-k+1} x^{k-1} \]
Therefore, we can write:
- Second term \( T_2 = \binom{n}{1} a^{n-1} x = 240 \)
- Third term \( T_3 = \binom{n}{2} a^{n-2} x^2 = 720 \)
- Fourth term \( T_4 = \binom{n}{3} a^{n-3} x^3 = 1080 \)

2. **Write the Equations**:
From the terms, we can set up the following equations:
\[ \text{(1)} \quad n a^{n-1} x = 240 \]
\[ \text{(2)} \quad \frac{n(n-1)}{2} a^{n-2} x^2 = 720 \]
\[ \text{(3)} \quad \frac{n(n-1)(n-2)}{6} a^{n-3} x^3 = 1080 \]

3. **Divide the Equations**:
To eliminate \( a \) and \( x \), we can divide the equations:
- Dividing (2) by (1):
\[ \frac{\frac{n(n-1)}{2} a^{n-2} x^2}{n a^{n-1} x} = \frac{720}{240} \]
Simplifying gives:
\[ \frac{(n-1)x}{2a} = 3 \quad \Rightarrow \quad \frac{a}{x} = \frac{(n-1)}{6} \quad \text{(4)} \]

- Dividing (3) by (2):
\[ \frac{\frac{n(n-1)(n-2)}{6} a^{n-3} x^3}{\frac{n(n-1)}{2} a^{n-2} x^2} = \frac{1080}{720} \]
Simplifying gives:
\[ \frac{(n-2)x}{3a} = \frac{3}{2} \quad \Rightarrow \quad \frac{a}{x} = \frac{2(n-2)}{9} \quad \text{(5)} \]

4. **Equate the Two Expressions for \( \frac{a}{x} \)**:
From equations (4) and (5):
\[ \frac{(n-1)}{6} = \frac{2(n-2)}{9} \]
Cross-multiplying gives:
\[ 9(n-1) = 12(n-2) \]
Expanding and simplifying:
\[ 9n - 9 = 12n - 24 \quad \Rightarrow \quad 3n = 15 \quad \Rightarrow \quad n = 5 \]

5. **Substitute \( n \) Back**:
Substitute \( n = 5 \) back into either equation (4) or (5) to find \( \frac{a}{x} \):
Using (4):
\[ \frac{a}{x} = \frac{5-1}{6} = \frac{4}{6} = \frac{2}{3} \]
Thus, \( a = \frac{2}{3}x \).

6. **Substitute \( a \) in Equation (1)**:
Substitute \( a \) into equation (1):
\[ 5 \left(\frac{2}{3}x\right)^{4} x = 240 \]
Simplifying:
\[ 5 \cdot \frac{16}{81} x^5 = 240 \quad \Rightarrow \quad \frac{80}{81} x^5 = 240 \quad \Rightarrow \quad x^5 = 240 \cdot \frac{81}{80} \]
\[ x^5 = 243 \quad \Rightarrow \quad x = 3 \]

7. **Find \( a \)**:
Substitute \( x = 3 \) back to find \( a \):
\[ a = \frac{2}{3} \cdot 3 = 2 \]

### Final Values:
Thus, we have:
\[ a = 2, \quad x = 3, \quad n = 5 \]