Arithmetic Sequence
Functions & Advanced AlgebraAn arithmetic sequence is a list of numbers where the difference between consecutive terms is always the same constant.
Formula
a_n = a_1 + (n-1)d
Definition
An arithmetic sequence is a pattern of numbers where you add the same amount each time to get the next number.
Example
$3, 7, 11, 15, 19, \ldots$ is arithmetic because you add $4$ each time. Also: $20, 15, 10, 5, 0, -5, \ldots$ is arithmetic because you subtract $5$ (add $-5$) each time.
Key Insight
An arithmetic sequence is like climbing stairs of equal height: each step (term) is the same distance above the last. The constant step size is called the common difference.
Definition
An arithmetic sequence has a constant common difference $d$ between consecutive terms: $a_{n+1} - a_n = d$. The explicit formula is $a_n = a_1 + (n - 1)d$, where $a_1$ is the first term and $n$ is the term number.
Example
Sequence: $5, 9, 13, 17, \ldots$ Here $a_1 = 5$, $d = 4$. The $10$th term: $a_{10} = 5 + (10-1)(4) = 5 + 36 = 41$. The $50$th term: $a_{50} = 5 + 49(4) = 201$.
Key Insight
Arithmetic sequences are linear functions of $n$. The explicit formula $a_n = a_1 + (n-1)d$ is the equation of a line with slope $d$ and $y$-intercept $a_1 - d$.
Definition
An arithmetic sequence is a sequence $\{a_n\}$ satisfying $a_n = a_1 + (n-1)d$ for all $n \ge 1$. It is equivalent to a linear function restricted to positive integers. The sequence defines an arithmetic progression in $\mathbb{Z}$ (or any ring), and its partial sums form an arithmetic series with closed-form sum $S_n = n/2 \cdot (2a_1 + (n-1)d)$.
Example
Dirichlet's theorem states that any arithmetic progression $a, a+d, a+2d, \ldots$ with $\gcd(a,d) = 1$ contains infinitely many primes, showing the deep number-theoretic importance of arithmetic sequences.
Key Insight
In abstract algebra, arithmetic sequences are cosets of the subgroup $d\mathbb{Z}$ in $\mathbb{Z}$. Their structure under modular arithmetic is central to the Chinese Remainder Theorem and cryptographic constructions.