Given:
The first term of the arithmetic progression (AP) is 1.

To find:
The common difference of the AP.

Solution:

Step 1: Understanding the problem:
An arithmetic progression is a sequence of numbers in which the difference between any two consecutive terms is constant. The first term of the AP is given as 1, and we need to find the common difference.

Step 2: Representing the AP:
Let's represent the arithmetic progression as {a, a + d, a + 2d, a + 3d, ...}, where 'a' is the first term and 'd' is the common difference.

Step 3: Finding the sum of the first four terms:
The sum of the first four terms of an AP can be calculated using the formula:
Sum = 4/2 * (2a + (4-1)d)

Substituting the given values, we have:
1/3 * Sum = 4/2 * (2 * 1 + (4-1) * d)

Simplifying, we get:
1/3 * Sum = 4/2 * (2 + 3d)

Step 4: Finding the sum of the next four terms:
The sum of the next four terms of an AP can also be calculated using the same formula:
Sum = 4/2 * (2(a + 4d) + (4-1)d)

Simplifying, we have:
1/3 * Sum = 4/2 * (2(a + 4d) + 3d)

Step 5: Equating the two sums:
We are given that the sum of the first four terms is one-third of the sum of the next four terms. So, we can equate the two sums.

4/2 * (2 + 3d) = 4/2 * (2(a + 4d) + 3d)

Cancelling out the common terms and simplifying, we get:
2 + 3d = 2(a + 4d) + 3d

Step 6: Solving the equation:
Expanding and simplifying the equation, we have:
2 + 3d = 2a + 8d + 3d

Combining like terms, we get:
2 + 3d = 2a + 11d

Rearranging the terms, we have:
2a = 2 + 3d - 11d

Simplifying further, we get:
2a = 2 - 8d

Dividing by 2, we have:
a = 1 - 4d

Step 7: Finding the common difference:
Since the first term 'a' is 1, we can substitute this value into the equation:
1 = 1 - 4d

Rearranging the terms, we have:
4d = 0

Dividing by 4, we get:
d = 0

Step 8: Conclusion:
The common difference of the arithmetic progression is 0.

Explanation:
Since the common difference is 0, it means that all the terms in the AP