Final answer:
Without additional information or context, we can only set up the equations based on the arithmetic progression (AP) but cannot solve for the sum of the first six terms given the current constraints.
Step-by-step explanation:
In an arithmetic progression (AP), if the sum of the first and fifth terms is 26 and the product of the second and fourth terms is 160, we can establish the relationship using the first term, a, and the common difference, d.
Let the first term be a and the common difference be d. Then the terms are as follows:
1st term: a
2nd term: a + d
3rd term: a + 2d
4th term: a + 3d
5th term: a + 4d
From the given, we have two equations:
a + (a + 4d) = 26
=> 2a + 4d = 26
and
(a + d) * (a + 3d) = 160
Once we find the values for a and d, we can calculate the sum of the first six terms of the AP by summing up these terms:
S6 = a + (a + d) + (a + 2d) + (a + 3d) + (a + 4d) + (a + 5d)
= 6a + 15d
Since the problem does not provide adequate information to solve for a and d, we cannot provide the exact value for S6 without further details or context.