The sum of first six terms of an arithmetic progression is 42. The ratio of its 10th term to its 30th term is 1: 3. The thirteenth term of the arithmetic progression is:

Question

Accepted Answer

To solve this problem, we need to determine the 13th term of an arithmetic progression (AP) given certain conditions. Let's break down the information provided and use it to find the correct answer, which is Option 2: 26.

### Explanation:

1. **Sum of the First Six Terms:**
   The sum of the first six terms of an AP is given by the formula:
   $$
   S_n = \frac{n}{2} \times (2a + (n-1)d)
   $$
   For the first six terms ($n = 6$), the sum is 42:
   $$
   \frac{6}{2} \times (2a + 5d) = 42
   $$
   Simplifying, we get:
   $$
   3 \times (2a + 5d) = 42 \quad \Rightarrow \quad 2a + 5d = 14 \quad \text{(Equation 1)}
   $$

2. **Ratio of the 10th Term to the 30th Term:**
   The nth term of an AP is given by:
   $$
   a_n = a + (n-1)d
   $$
   For the 10th term:
   $$
   a_{10} = a + 9d
   $$
   For the 30th term:
   $$
   a_{30} = a + 29d
   $$
   The ratio of the 10th term to the 30th term is 1:3:
   $$
   \frac{a + 9d}{a + 29d} = \frac{1}{3}
   $$
   Cross-multiplying gives:
   $$
   3(a + 9d) = a + 29d
   $$
   Simplifying, we get:
   $$
   3a + 27d = a + 29d \quad \Rightarrow \quad 2a = 2d \quad \Rightarrow \quad a = d \quad \text{(Equation 2)}
   $$

3. **Finding the 13th Term:**
   Using Equation 2 ($a = d$) in Equation 1:
   $$
   2a + 5d = 14 \quad \Rightarrow \quad 2d + 5d = 14 \quad \Rightarrow \quad 7d = 14 \quad \Rightarrow \quad d = 2
   $$
   Since $a = d$, we have $a = 2$.

Now, the 13th term is:
   $$
   a_{13} = a + 12d = 2 + 12 \times 2 = 2 + 24 = 26
   $$

### Conclusion:

- **Option 2 (26) is correct** because the calculations show that the 13th term of the AP is indeed 26.

- **Why Other Options are Incorrect:**
  - **Option 1 (62):** This would imply a much larger common difference or first term, which contradicts the given conditions.
  - **Option 3 (28):** This value does not satisfy the derived equations and conditions.
  - **Option 4 (82):** This is far too large given the constraints of the problem.

Thus, the correct answer is Option 2: 26, as supported by the calculations and logical deductions based on the given conditions.