Given Information:
- The 6th term of an arithmetic progression (AP) is one more than twice the third term.
- The sum of the 4th and 5th term is five times the second term.

Let's solve the problem step by step:

Step 1: Understanding the Problem
- We are given an arithmetic progression (AP) with unknown terms.
- We need to find the 10th term of the AP.

Step 2: Setting up Equations
- Let's assume that the first term of the AP is 'a' and the common difference is 'd'.
- The third term of the AP can be written as a + 2d.
- The sixth term of the AP can be written as a + 5d.
- According to the first condition, the sixth term is one more than twice the third term, so we can write the equation as:
a + 5d = 2(a + 2d) + 1
- Simplifying the equation, we get:
a + 5d = 2a + 4d + 1
a - 2a = 4d - 5d + 1
-a = -d + 1
a = d - 1

- The second term of the AP can be written as a + d.
- The fourth term of the AP can be written as a + 3d.
- The fifth term of the AP can be written as a + 4d.
- According to the second condition, the sum of the 4th and 5th term is five times the second term, so we can write the equation as:
(a + 3d) + (a + 4d) = 5(a + d)
2a + 7d = 5a + 5d
2a - 5a = 5d - 7d
-3a = -2d
a = (2/3)d

Step 3: Finding the 10th Term
- We now have two equations:
a = d - 1
a = (2/3)d

- Solving these equations simultaneously, we can find the value of 'd' and 'a'.
- Substitute the value of 'a' in the second equation:
(2/3)d = d - 1
2d = 3d - 3
-d = -3
d = 3

- Substitute the value of 'd' in the first equation:
a = 3 - 1
a = 2

- Now, we know that the first term 'a' is 2 and the common difference 'd' is 3.
- The 10th term of the AP can be written as a + 9d:
10th term = 2 + 9 * 3
= 2 + 27
= 29

Step 4: Conclusion
- The 10th term of the arithmetic progression (AP) is 29.

Answer