Problem:
If 5 times the fifth term of an arithmetic progression (A.P) is equal to 8 times the eighth term of that A.P, then what is the thirteenth term of the A.P?

Solution:

Step 1: Understanding the problem
We are given that 5 times the fifth term of an A.P is equal to 8 times the eighth term. We need to find the thirteenth term of the A.P.

Step 2: Understanding arithmetic progression
An arithmetic progression is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is denoted by 'd'.

Step 3: Representing the A.P
Let's assume that the first term of the A.P is 'a' and the common difference is 'd'. The fifth term of the A.P can be represented as 'a + 4d' and the eighth term can be represented as 'a + 7d'.

Step 4: Setting up the equation
According to the given information, 5 times the fifth term is equal to 8 times the eighth term. Mathematically, this can be represented as:
5(a + 4d) = 8(a + 7d)

Step 5: Solving the equation
Let's solve the equation to find the values of 'a' and 'd'.

5(a + 4d) = 8(a + 7d)
5a + 20d = 8a + 56d
20d - 56d = 8a - 5a
-36d = 3a

Step 6: Finding the thirteenth term
Now that we have the relationship between 'a' and 'd', we can find the value of 'd' in terms of 'a'. Dividing both sides of the equation by 3, we get:
d = -a/12

To find the thirteenth term, we can substitute the values of 'a' and 'd' in the equation for the nth term of an A.P:
T(n) = a + (n - 1)d

Substituting n = 13, a = a, and d = -a/12, we get:
T(13) = a + (13 - 1)(-a/12)
T(13) = a - (12/12)a
T(13) = a - a
T(13) = 0

Step 7: Conclusion
The thirteenth term of the arithmetic progression is 0.