Find the nth term and sum of arithmetic sequences instantly
The nth term of an arithmetic sequence is found using a_n = a_1 + (n-1)d, where a_1 is the first term and d is the common difference. The sum of the first n terms is S_n = n/2 * (2a_1 + (n-1)d). These formulas let you find any term or partial sum without listing every element.
Arithmetic sequences appear everywhere -- from calculating monthly savings growth to predicting patterns in data. Understanding them helps with financial planning, scheduling, and solving problems in physics and engineering where quantities change at a constant rate.
The nth term tells you the exact value at any position in the sequence. The sum gives you the total of all terms up to that position. The sequence preview shows the first few terms so you can verify the pattern looks correct.
Always identify the common difference first by subtracting consecutive terms. Double-check that the difference is truly constant -- if it varies, you may have a geometric or other type of sequence instead. Use the sum formula for large sequences rather than adding terms individually.
An arithmetic sequence is a series of numbers where the difference between consecutive terms is constant. For example, 2, 5, 8, 11, 14 is an arithmetic sequence with a common difference of 3. Each term is found by adding the common difference to the previous term.
The common difference is found by subtracting any term from the term that follows it. For example, in the sequence 4, 9, 14, 19, the common difference is 9 - 4 = 5. If the difference between consecutive terms is not constant, the sequence is not arithmetic.
The sum of the first n terms of an arithmetic series is S_n = n/2 * (2a_1 + (n-1)d), where a_1 is the first term, d is the common difference, and n is the number of terms. An equivalent formula is S_n = n/2 * (a_1 + a_n), where a_n is the last term.