Asked by Alto saxaphone
what is an arithmetic sequence
All Answers 1
Answered by
GPT-5 mini
AI
An arithmetic sequence (aka arithmetic progression, AP) is a sequence of numbers in which the difference between any two consecutive terms is constant. That constant is called the common difference.
Key facts and formulas
- Common difference: d = a_{n+1} − a_n (same for every n).
- nth term: a_n = a_1 + (n − 1)d (equivalently a_n = a_k + (n − k)d).
- Sum of the first n terms: S_n = n/2 · (2a_1 + (n − 1)d) = n · (a_1 + a_n)/2.
- To test if a sequence is arithmetic: check that all consecutive differences are equal.
- If d = 0 the sequence is constant. If d ≠ 0 an infinite arithmetic sequence is unbounded (terms go to ±∞).
Examples
- 2, 5, 8, 11, ... is arithmetic with d = 3 and a_n = 2 + (n − 1)3.
- 10, 7, 4, 1, ... has d = −3 and a_n = 10 + (n − 1)(−3).
If you want, I can show how to find a specific term, the common difference from two given terms, or the sum of the first n terms for an example.
Key facts and formulas
- Common difference: d = a_{n+1} − a_n (same for every n).
- nth term: a_n = a_1 + (n − 1)d (equivalently a_n = a_k + (n − k)d).
- Sum of the first n terms: S_n = n/2 · (2a_1 + (n − 1)d) = n · (a_1 + a_n)/2.
- To test if a sequence is arithmetic: check that all consecutive differences are equal.
- If d = 0 the sequence is constant. If d ≠ 0 an infinite arithmetic sequence is unbounded (terms go to ±∞).
Examples
- 2, 5, 8, 11, ... is arithmetic with d = 3 and a_n = 2 + (n − 1)3.
- 10, 7, 4, 1, ... has d = −3 and a_n = 10 + (n − 1)(−3).
If you want, I can show how to find a specific term, the common difference from two given terms, or the sum of the first n terms for an example.
Submit Your Answer
We prioritize human answers over AI answers.
If you are human, and you can answer this question, please submit your answer.