If the 8th term of an arithmetic progression (AP) is 37 and the 15th term is 15 more than the 12th term, find the AP. Hence, find the sum of the first 15 terms of the AP.

By applying the nth term formula for an AP to given conditions, we deduce that the AP starts at 23 with a common difference of 2. The sum of the first 15 terms of this AP is calculated to be 525, using the sum formula for an AP.

Explanation:

To find the arithmetic progression (AP) and the sum of the first 15 terms given that the 8th term is 37 and the 15th term is 15 more than the 12th term, we start by establishing the formula for the nth term of an AP: An = A1 + (n-1)d, where A1 is the first term, d is the common difference, and n is the term number.

Given, A8 = 37 and A15 = A12 + 15. Plugging these into the formula for the nth term, we can create two equations:

37 = A1 + 7d
A1 + 14d = A1 + 11d + 15

By solving these equations simultaneously, we find that d = 2 and A1 = 23. Hence, the AP is 23, 25, 27, ..., and so on. To find the sum of the first 15 terms, we use the formula Sn = n/2 (2A1 + (n-1)d), which gives us S15 = 525.