Find the first term of an arithmetic progression (AP) and the common difference in the series, given that the sum of the first 6 terms of the AP is 18. If the sum of the first 22 terms of the AP is 6, find the common difference.

To find the first term and common difference of the AP, two equations are derived from the sums of the first 6 and 22 terms. By simplifying and solving these equations simultaneously, the first term and common difference can be calculated.

Explanation:

To find the first term of an arithmetic progression (AP) and the common difference, we must utilize the formula for the sum of the first n terms of an AP, which is Sn = n/2 (2a + (n - 1)d), where Sn is the sum of the first n terms, a is the first term, and d is the common difference.

Given that the sum of the first 6 terms (S6) is 18, and the sum of the first 22 terms (S22) is 6, we can set up two equations:

18 = 6/2 (2a + 5d)
6 = 22/2 (2a + 21d)

Solving these two equations simultaneously, we can find the values for a and d.

Step 1: Simplify the equations:

18 = 3 (2a + 5d)
6 = 11 (2a + 21d)

Step 2: Divide both equations by their coefficients to isolate the terms with a and d:

6 = 2a + 5d
6/11 = 2a + 21d

Step 3: Multiply both sides of the second equation by 11 to make the coefficients of the a terms equal:

6 = 2a + 5d
6 = 22a + 231d

By subtracting the second equation from the first, we find the common difference in the series.