Let the sum of the first n terms of a non-constant arithmetic progression a₁, a₂, a₃, ..., be given. If A is a constant and d is the common difference of this arithmetic progression, then the ordered pair (d, a₁) is equal to

In an arithmetic progression (AP), the sum of the first n terms can be expressed using the formula:

Sum of the First n Terms

The sum \( S_n \) of the first n terms of an AP is given by:

S_n = \frac{n}{2} (2a_1 + (n-1)d)

Understanding the Variables

• a₁: The first term of the AP.
• d: The common difference between consecutive terms.
• n: The number of terms being summed.

Finding the Ordered Pair (d, a₁)

To find the ordered pair (d, a₁), we can rearrange the sum formula. If we know the sum \( S_n \) and the number of terms n, we can express a₁ in terms of d:

a₁ = \frac{2S_n}{n} - \frac{(n-1)d}{2}

Conclusion

Thus, the ordered pair (d, a₁) can be derived from the known values of the sum \( S_n \) and the number of terms n. The specific values of d and a₁ will depend on the context of the problem and the given sum.