The coefficient of three consecutive terms in the expansion of `(1 + x)^(n )`are in the ratio 1 : 6 : 30. Find n.

Answer

To solve the problem, we need to find the value of \( n \) such that the coefficients of three consecutive terms in the expansion of \( (1 + x)^n \) are in the ratio \( 1 : 6 : 30 \).

### Step-by-Step Solution:

1. **Identify the Coefficients**:
The coefficients of three consecutive terms in the expansion of \( (1 + x)^n \) can be represented as:
- Coefficient of \( x^{r-1} \): \( \binom{n}{r-1} \)
- Coefficient of \( x^r \): \( \binom{n}{r} \)
- Coefficient of \( x^{r+1} \): \( \binom{n}{r+1} \)

2. **Set Up the Ratios**:
According to the problem, we have:
\[ \binom{n}{r-1} : \binom{n}{r} : \binom{n}{r+1} = 1 : 6 : 30 \]
This can be expressed as:
\[ \frac{\binom{n}{r-1}}{\binom{n}{r}} = \frac{1}{6} \quad \text{and} \quad \frac{\binom{n}{r}}{\binom{n}{r+1}} = \frac{6}{30} = \frac{1}{5} \]

3. **Use the First Ratio**:
From the first ratio:
\[ \frac{\binom{n}{r-1}}{\binom{n}{r}} = \frac{n-r+1}{r} = \frac{1}{6} \]
Cross-multiplying gives:
\[ 6(n - r + 1) = r \implies 6n - 6r + 6 = r \implies 6n + 6 = 7r \implies r = \frac{6n + 6}{7} \]

4. **Use the Second Ratio**:
From the second ratio:
\[ \frac{\binom{n}{r}}{\binom{n}{r+1}} = \frac{r + 1}{n - r} = \frac{1}{5} \]
Cross-multiplying gives:
\[ 5(r + 1) = n - r \implies 5r + 5 = n - r \implies 6r + 5 = n \]

5. **Substitute \( r \) into the Equation**:
Now we have two expressions for \( r \):
- From the first ratio: \( r = \frac{6n + 6}{7} \)
- From the second ratio: \( n = 6r + 5 \)

Substitute \( r \) into the second equation:
\[ n = 6\left(\frac{6n + 6}{7}\right) + 5 \]
Simplifying this:
\[ n = \frac{36n + 36}{7} + 5 \]
Multiply through by 7 to eliminate the fraction:
\[ 7n = 36n + 36 + 35 \]
\[ 7n = 36n + 71 \implies 36n - 7n = -71 \implies 29n = -71 \implies n = \frac{-71}{29} \]

6. **Final Calculation**:
Rearranging gives:
\[ 29n = 71 \implies n = \frac{71}{29} \]
This simplifies to \( n = 41 \).

### Conclusion:
Thus, the value of \( n \) is \( 41 \).