The 3rd term of an Arithmetic Progression is 10 more than the first term while the fifth term is 15 more than the second term. Find the sum of the 8th and 15th terms of the Arithmetic Progression if the 7th term is seven times the first term.

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"The 3rd term of an Arithmetic Progression is 10 more than the first term" ----> a+2d = a+10

2d=10
d = 5
"the 7th term is seven times the first term"
--> a+6d = 7a
6d = 6a
a= d = 5

sum of 8ht and 15th terms
= a+7d + a + 14d
= 2a + 21d
= 10 + 105 = 115

Why don't we start by figuring out the values of the first term and the common difference?

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.

To solve this question, we first need to find the values of the first term (a) and the common difference (d) of the arithmetic progression.

Let's start by finding the values of the first term (a) and the common difference (d) using the given information:

Given:
- The 3rd term is 10 more than the first term. So, a + 2d = a + 10. => 2d = 10. => d = 10/2 = 5.
- The fifth term is 15 more than the second term. So, a + 4d = (a + 2d) + 15. => 4d - 2d = 15. => 2d = 15. => d = 15/2.

Now, we have the value of common difference (d) as 5, which is equal to 15/2. This means there is a contradiction in the given information, so it is not possible to find the first term and the common difference of the arithmetic progression.

Therefore, we cannot proceed to find the sum of the 8th and 15th terms of the arithmetic progression.

If you have any other questions, feel free to ask!