Answer: Let's call the first term of the arithmetic progression "a" and the common difference "d". Then we can use the following formulas to find the second, fifth, and seventh terms:
- The second term is a + d.
- The fifth term is a + 4d.
- The seventh term is a + 6d.
We are given that the sum of the second and seventh terms is 25:
(a + d) + (a + 6d) = 25
Simplifying this equation, we get:
2a + 7d = 25 ...(1)
We are also given that the fifth term is 15:
a + 4d = 15 ...(2)
Now we can solve for the common difference "d" by using equations (1) and (2) to form a system of two equations in two variables. One way to do this is to solve equation (2) for "a" in terms of "d", and then substitute that expression for "a" into equation (1):
a = 15 - 4d ...(3)
Substituting equation (3) into equation (1), we get:
2(15 - 4d) + 7d = 25
Simplifying this equation, we get:
30 - 8d + 7d = 25
Solving for "d", we get:
d = -5
Therefore, the common difference is -5.
Explanation: