Let's solve this step-by-step.
Step 1: Understand the problem.
We are given that the 3rd term of an arithmetic progression (AP) is 10 more than the first term, and the 5th term is 15 more than the second term. Additionally, the 7th term is seven times the first term. We need to find the sum of the 8th and 15th terms of the AP.
Step 2: Define the arithmetic progression.
Let's assume that the first term of the AP is 'a' and the common difference is 'd'.
Step 3: Find the third term.
The third term is given to be 10 more than the first term, so we have:
a + 2d = a + 10.
Step 4: Find the fifth term.
The fifth term is given to be 15 more than the second term, so we have:
a + 4d = (a + d) + 15.
Step 5: Find the seventh term.
The seventh term is given to be seven times the first term, so we have:
a + 6d = 7a.
Step 6: Solve the equations.
From Step 3, we get:
2d = 10,
d = 5.
From Step 4, we substituted the value of 'd' and obtained:
a + 20 = a + 15,
5 = 15.
This means that the given information is inconsistent and doesn't satisfy the conditions of an arithmetic progression. Since we cannot proceed with this information, we cannot calculate the sum of the 8th and 15th terms.
Therefore, we cannot find the sum of the 8th and 15th terms of the arithmetic progression given the information provided.