**Introduction**

The Lyman series refers to the set of spectral lines that are produced when an electron transitions from a higher energy level to the first energy level (n=1) in a hydrogen-like atom. These transitions result in the emission of photons with specific wavelengths. The Lyman series is part of the broader hydrogen spectrum and is characterized by higher energy levels compared to other series, such as the Balmer series.

**Explanation of the Ratios**

The ratio of the difference in wavelength between the first and second lines of the Lyman series to the difference in wavelength between the second and third lines can be determined by considering the energy levels involved in the transitions.

1. **Difference in wavelength between first and second lines:**
- The first line in the Lyman series corresponds to the transition from the second energy level (n=2) to the first energy level (n=1). This transition is associated with the emission of a photon with a specific wavelength, λ₁.
- The second line corresponds to the transition from the third energy level (n=3) to the first energy level (n=1). This transition results in the emission of a photon with a different wavelength, λ₂.
- The difference in wavelength between these two lines can be calculated as Δλ₁₂ = λ₂ - λ₁.

2. **Difference in wavelength between second and third lines:**
- The second line of the Lyman series involves the transition from the third energy level (n=3) to the first energy level (n=1), resulting in the emission of a photon with wavelength λ₂.
- The third line corresponds to the transition from the fourth energy level (n=4) to the first energy level (n=1), resulting in the emission of a photon with a different wavelength, λ₃.
- The difference in wavelength between these two lines can be calculated as Δλ₂₃ = λ₃ - λ₂.

**Calculating the Ratio**

To find the ratio of the two differences in wavelength, we can express the wavelengths in terms of the energy levels involved and use the Rydberg formula:

1. The wavelength λ₁ can be expressed as λ₁ = R(1/1² - 1/2²), where R is the Rydberg constant.
2. The wavelength λ₂ can be expressed as λ₂ = R(1/1² - 1/3²).
3. The wavelength λ₃ can be expressed as λ₃ = R(1/1² - 1/4²).

Now, substituting these expressions into the differences in wavelength:

1. Δλ₁₂ = λ₂ - λ₁ = R(1/1² - 1/3²) - R(1/1² - 1/2²) = R(1/9 - 1/1) + R(1/4 - 1/1) = R(8/9 - 3/4).
2. Δλ₂₃ = λ₃ - λ₂ = R(1/1² - 1/4²) - R(1/1² - 1/3²) = R(1/16 - 1/1) + R(1/3 - 1/1) = R(15/16 - 2/3).

Finally, calculating the ratio:

Ratio = Δ

2.5:1