Problem:
The sum of the 3rd term and 7th term of an arithmetic progression (AP) is 6 and their product is 8. We need to find the sum of the first 16 terms.

Solution:

Step 1: Understanding the Problem
Let's assume the first term of the AP is 'a' and the common difference is 'd'. We are given that the sum of the 3rd term and 7th term is 6 and their product is 8. We need to find the sum of the first 16 terms.

Step 2: Expressing the given information mathematically
We can express the 3rd term and 7th term of the AP using the formulas:
3rd term = a + 2d
7th term = a + 6d

According to the problem, the sum of the 3rd term and 7th term is 6, so we have the equation:
(a + 2d) + (a + 6d) = 6

The product of the 3rd term and 7th term is given as 8, so we have the equation:
(a + 2d)(a + 6d) = 8

Step 3: Solving the equations
Let's solve the two equations simultaneously to find the values of 'a' and 'd'.

Expanding the second equation, we get:
a^2 + 8ad + 12d^2 = 8

Rearranging the first equation, we get:
2a + 8d = 6

We can solve these equations using substitution or elimination method. Let's use the elimination method.

Multiplying the first equation by 2, we get:
4a + 16d = 12

Subtracting this equation from the second equation, we eliminate 'a':
(2a + 8d) - (4a + 16d) = 6 - 12
-2a - 8d = -6

Simplifying, we have:
-2a - 8d = -6

Dividing the equation by -2, we get:
a + 4d = 3 ----(1)

Now, we can solve equations (1) and (2) simultaneously to find the values of 'a' and 'd'.

Step 4: Finding the values of 'a' and 'd'
Solving equations (1) and (2) simultaneously, we get:
a + 4d = 3 ----(1)
2a + 8d = 6 ----(2)

Multiplying equation (1) by 2, we get:
2a + 8d = 6

Comparing this equation with equation (2), we can see that they are the same. This means that equation (1) is redundant and doesn't provide any additional information. Therefore, we cannot determine unique values for 'a' and 'd' based on the given information.

Step 5: Conclusion
Since we cannot determine the values of 'a' and 'd', we cannot find the sum of the first 16 terms of the AP.