In the spectrum of hydrogen, the ratio of the longest wavelength in the Lyman series to the longest wavelength in the Balmer series is `(x)/(27)`. Find `x`.

Answer

To solve the problem, we need to find the ratio of the longest wavelength in the Lyman series to the longest wavelength in the Balmer series for the hydrogen spectrum.

### Step-by-Step Solution:

1. **Identify the longest wavelength in the Lyman series**:
- The longest wavelength in the Lyman series occurs when the electron transitions from the n=2 level to the n=1 level.
- The formula for the wavelength (λ) in the hydrogen spectrum is given by:
\[ \frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]
- For the Lyman series (n1 = 1, n2 = 2):
\[ \frac{1}{\lambda_1} = R \left( \frac{1}{1^2} - \frac{1}{2^2} \right) = R \left( 1 - \frac{1}{4} \right) = R \left( \frac{3}{4} \right) \]
- Therefore, we can express λ1 as:
\[ \lambda_1 = \frac{4}{3R} \]

2. **Identify the longest wavelength in the Balmer series**:
- The longest wavelength in the Balmer series occurs when the electron transitions from the n=3 level to the n=2 level.
- For the Balmer series (n1 = 2, n2 = 3):
\[ \frac{1}{\lambda_2} = R \left( \frac{1}{2^2} - \frac{1}{3^2} \right) = R \left( \frac{1}{4} - \frac{1}{9} \right) \]
- To simplify this, find a common denominator (36):
\[ \frac{1}{\lambda_2} = R \left( \frac{9}{36} - \frac{4}{36} \right) = R \left( \frac{5}{36} \right) \]
- Therefore, we can express λ2 as:
\[ \lambda_2 = \frac{36}{5R} \]

3. **Calculate the ratio of the longest wavelengths**:
- Now, we need to find the ratio of λ1 to λ2:
\[ \frac{\lambda_1}{\lambda_2} = \frac{\frac{4}{3R}}{\frac{36}{5R}} = \frac{4}{3R} \cdot \frac{5R}{36} = \frac{4 \cdot 5}{3 \cdot 36} = \frac{20}{108} = \frac{5}{27} \]

4. **Relate the ratio to the given expression**:
- According to the problem, this ratio is given as:
\[ \frac{x}{27} \]
- From our calculation, we have:
\[ \frac{5}{27} = \frac{x}{27} \]
- Thus, we can conclude that:
\[ x = 5 \]

### Final Answer:
The value of \( x \) is \( 5 \).