The sum of the first six terms of an AP is 42. The ratio of its 10th term to its 13th term is 1:3. Calculate the first and the thirteenth terms of the AP.

To find the first and thirteenth term of an arithmetic progression (AP), we can use the sum of the first six terms and the ratio between the tenth and thirteenth term. By solving the equations, we can find the values of the first term and the thirteenth term.

Explanation:

The sum of the first six terms of an arithmetic progression (AP) is given as 42. Let's assume the first term of the AP is 'a' and the common difference is 'd'.

Using the formula for the sum of an AP, we have:

6/2 * (2a + (6-1)d) = 42

3(2a + 5d) = 42

2a + 5d = 14

Now, we are given that the ratio of the 10th term to the 13th term is 1:3.

Using the formula for the nth term of an AP, we have:

The 10th term: a + 9d

The 13th term: a + 12d

According to the given ratio, we can write:

(a + 9d) / (a + 12d) = 1/3

3(a + 9d) = a + 12d

3a + 27d = a + 12d

2a - 15d = 0

By solving these two equations simultaneously, you can find the values of the first term (a) and the thirteenth term (a + 12d).