The third term of an arithmetic progression (AP) is 7. The seventh term exceeds three times the third term by the sum of the first term, the common difference, and the sum of the first 20 terms. Find the common difference of the AP.

To find the common difference of the AP, we use the formula for the nth term and the sum of the first n terms. By substituting known values from the conditions provided, we can set up an equation and solve for the common difference.

Explanation:

The given problem involves finding the common difference of an arithmetic progression (AP) given some conditions. From the information provided:

The third term of the AP is 7, represented as T3 = 7.
The seventh term exceeds three times the third term by the sum of the first term, the common difference, and the sum of the first 20 terms.

The formula for the nth term of an AP is: Tn = a + (n - 1)d, where a is the first term, d is the common difference, and n is the term number.

Applying this to the third term gives us T3 = a + 2d = 7.

The sum of the first n terms of an AP is given by: Sn = ½n(2a + (n - 1)d). Thus, the sum of the first 20 terms is: S20 = 10(2a + 19d).

The condition given can be represented as T7 = 3T3 + a + d + S20. Substituting known values and simplifying, we can solve for the common difference, d.