Arithmetic Sequence Calculator

An arithmetic sequence is a list of numbers where the difference between any two consecutive terms is always constant. This constant value is known as the common difference, denoted as d. To find any specific term in the sequence (the nth term, or a_n), the calculator uses the explicit formula: a_n = a₁ + d(n - 1), where a₁ is the first term and n is the position of the term. If you need to find a term when you only know a different term (like a_m), the generalized formula applies: a_n = a_m + d(n - m). Additionally, finding the sum of a finite arithmetic series is achieved using the formula for the sum of the first n terms.

Imagine you are given an arithmetic sequence where the first term (a₁) is 4, and the common difference (d) is 3. You want to find the 10th term (a₁₀) and the sum of the first 10 terms. First, apply the explicit term formula: a₁₀ = 4 + 3(10 - 1). Solving inside the parentheses gives 10 - 1 = 9. Next, multiply by the common difference: 3 × 9 = 27. Finally, add the first term: 4 + 27 = 31. The 10th term of your sequence is 31. To find the sum, you would aggregate the progression: 4, 7, 10, 13, 16, 19, 22, 25, 28, and 31, yielding a total sum of 175.

When analyzing a sequence, always verify at least three consecutive terms to confirm that the common difference remains strictly constant. Watch out for negative numbers when calculating the difference; subtracting a negative number is equivalent to adding its positive counterpart. Utilizing automated tools helps bypass tedious hand calculations, allowing you to focus on analyzing broader algebraic patterns and series behavior.