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The question in the image is about an arithmetic progression (समान्तर श्रेणी) where the sum of the first \(n\) terms, \(S_n\), is given, and the ratio of the sums of the first and fourth terms \(S_1 : S_4 = 1 : 10\). We are tasked with finding the ratio of the first term to the fourth term.

Let’s break it down step by step:

1. **Sum of first \(n\) terms** of an arithmetic progression:
   \[
   S_n = \frac{n}{2} [2a + (n-1)d]
   \]
   where \(a\) is the first term and \(d\) is the common difference.

2. Using the given ratio \(S_1 : S_4 = 1 : 10\), we can set up the sum equations for \(S_1\) and \(S_4\):
   - \(S_1 = a\)
   - \(S_4 = 2a + 3d\)

   The ratio becomes:
   \[
   \frac{S_1}{S_4} = \frac{1}{10} \quad \Rightarrow \quad \frac{a}{2a + 3d} = \frac{1}{10}
   \]

3. Solving for \(a\) and \(d\):
   \[
   10a = 2a + 3d
   \]
   \[
   8a = 3d \quad \Rightarrow \quad d = \frac{8a}{3}
   \]

4. The **fourth term** in the arithmetic progression is given by:
   \[
   T_4 = a + 3d = a + 3 \times \frac{8a}{3} = a + 8a = 9a
   \]

5. Therefore, the ratio of the first term to the fourth term is:
   \[
   \frac{T_1}{T_4} = \frac{a}{9a} = \frac{1}{9}
   \]

The correct option should be the one closest to this ratio. However, none of the options exactly match this value, but it seems there might be an approximation or error. Based on the logic, none of the provided options directly fit.

Would you like further details, or shall we adjust the logic? Also, here are some related questions you might find useful:

1. What is the formula for the sum of the first \(n\) terms of an arithmetic progression?
2. How is the common difference \(d\) calculated in an arithmetic progression?
3. What is the relationship between the sum and the terms of an arithmetic progression?
4. How can we derive the value of the \(n\)-th term in an arithmetic progression?
5. What is the significance of the ratio in arithmetic progressions?

**Tip:** Always double-check the final terms and sums when dealing with progression problems to avoid calculation mistakes.

The image contains a question related to an arithmetic progression where the sum S_n of the first n terms is known, and the ratio S_1 : S_4 = 1:10. We are asked to find the ratio of the first term to the fourth term.

Explore how to determine the ratio of the first term to the fourth term in an arithmetic progression where the sum of the first n terms (S_n) and the ratio of the sums of the first and fourth terms (S_1 : S_4 = 1:10) are given. This problem uses the fundamental formulas for sums and terms within arithmetic sequences.

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