Given information:

- The sum of the first fifteen terms of an arithmetic progression is 40 more than the sum of the first seven terms of the same arithmetic progression.

Let's assume that the first term of the arithmetic progression is 'a' and the common difference is 'd'.

Sum of the first fifteen terms:
S₁₅ = (15/2)(2a + (15-1)d) = 15a + 105d

Sum of the first seven terms:
S₇ = (7/2)(2a + (7-1)d) = 7a + 21d

According to the given information, the sum of the first fifteen terms is 40 more than the sum of the first seven terms. Mathematically, we can represent this as:

S₁₅ = S₇ + 40

Substituting the values of S₁₅ and S₇, we get:

15a + 105d = 7a + 21d + 40

Simplifying the equation, we get:

8a + 84d = 40

Dividing both sides of the equation by 4, we get:

2a + 21d = 10

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To find the sum of the first twenty-two terms of the arithmetic progression, we can use the formula for the sum of an arithmetic progression:

S₂₂ = (22/2)(2a + (22-1)d) = 11(2a + 21d) = 22a + 231d

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To find the value of S₂₂, we need to find the values of 'a' and 'd'. We can do this by solving the given equation:

2a + 21d = 10

Since there are infinitely many solutions to this equation, we need another equation to solve for 'a' and 'd'.

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We can use the equation for the sum of the first fifteen terms to find another equation:

15a + 105d = 7a + 21d + 40

Simplifying the equation, we get:

8a + 84d = 40

Dividing both sides of the equation by 4, we get:

2a + 21d = 10

This equation is the same as the one we found earlier, so we can use it to solve for 'a' and 'd'.

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Substituting the value of 2a + 21d from the equation 2a + 21d = 10 into the equation 2a + 21d = 10, we get:

10 = 10

This equation is true for all values of 'a' and 'd'. Therefore, there are infinitely many solutions to the given problem.

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Since there are infinitely many solutions, we cannot determine the exact value of S₂₂. However, we can determine the relationship between the sum of the first twenty-two terms and the sum of the first fifteen terms.

The sum of the first twenty-two terms is always greater than the sum of the first fifteen terms by a constant value of 22a + 231d. Therefore, the sum of the first twenty-two terms is always 231 greater than the sum of the first fifteen terms.

Hence, the sum of the first twenty-two terms of the arithmetic progression is 231 more than