Mathematics.

sequences and series

Arithmetic Sequences

Algebra II30 minDifficulty3 out of 10

You should know: sequences and series

Overview

An arithmetic sequence is a sequence in which each term differs from the previous one by a constant amount, called the common difference d. Because the increment is always the same, arithmetic sequences correspond to linear functions of the index n, and their sums (arithmetic series) have a clean closed-form formula discovered — according to legend — by a young Carl Friedrich Gauss.

Intuition

Picture a staircase where every step is exactly the same height. An arithmetic sequence is that staircase in number form: start at a₁, and add the same fixed amount d to get each next term. Because the steps are equal-sized, the n-th term is just the starting value plus (n-1) full steps of size d — no curvature, no compounding, just steady linear change.

Interactive Graph

Term value a + d*n as a continuous approximation

Loading visualization…

Formal Definition

Definition

An arithmetic sequence has a constant common difference d between consecutive terms:

an=a1+(n−1)da_n = a_1 + (n-1)d

a₁ is the first term, d is the common difference

Explicit formula
an=an−1+da_n = a_{n-1} + d
Recursive formula

Derivation

Gauss's trick for summing an arithmetic series: write the sum forwards and backwards, then add term by term.

Sn=a1+a2+⋯+anS_n = a_1 + a_2 + \cdots + a_n

Sum written forwards

Sn=an+an−1+⋯+a1S_n = a_n + a_{n-1} + \cdots + a_1

Same sum written backwards

2Sn=(a1+an)+(a2+an−1)+⋯+(an+a1)2S_n = (a_1+a_n) + (a_2+a_{n-1}) + \cdots + (a_n+a_1)

Add the two equations term by term

2Sn=n(a1+an)  ⟹  Sn=n2(a1+an)2S_n = n(a_1+a_n) \implies S_n = \frac{n}{2}(a_1+a_n)

Every one of the n pairs sums to the same value (a₁+aₙ), since the sequence is linear

Properties

Common difference

d=an−an−1 (constant for all n)d = a_{n} - a_{n-1} \text{ (constant for all } n\text{)}

Arithmetic series sum

Sn=n2(a1+an)=n2[2a1+(n−1)d]S_n = \frac{n}{2}(a_1 + a_n) = \frac{n}{2}\big[2a_1 + (n-1)d\big]

Arithmetic mean property

an=an−1+an+12 (each term is the average of its neighbors)a_n = \frac{a_{n-1}+a_{n+1}}{2} \text{ (each term is the average of its neighbors)}

Applications

Simple interest (not compound) grows an account by the same fixed dollar amount each period, an arithmetic sequence.

Worked Examples

  1. Identify a₁=4 and d=3, then apply the explicit formula.

    a10=4+(10−1)(3)=4+27=31a_{10} = 4 + (10-1)(3) = 4+27 = 31

Answer: a₁₀ = 31

Practice Problems

Difficulty 2/10

Find the common difference and the 8th term of the sequence 2, 5, 8, 11, ...

Difficulty 4/10

Find the sum of the first 50 positive integers.

Common Mistakes

Common Mistake

Using aₙ = a₁ + n·d instead of aₙ = a₁ + (n-1)d.

The first term a₁ requires ZERO steps of size d (it's the starting point), so the exponent/multiplier on d must be (n-1), not n. Check: a₁ = a₁+(1-1)d = a₁ ✓.

Common Mistake

Applying the arithmetic sum formula Sₙ=n/2(a₁+aₙ) to a sequence that isn't arithmetic.

This formula relies on every consecutive pair of terms having the same difference d. It gives wrong answers for geometric or other non-linear sequences — always verify the common difference is constant first.

Summary

  • An arithmetic sequence has a constant common difference d between consecutive terms: aₙ = a₁+(n-1)d.
  • The sum of the first n terms is Sₙ = (n/2)(a₁+aₙ), derived via Gauss's forward/backward pairing trick.
  • Each term is the arithmetic mean of its two neighbors.
  • Arithmetic sequences model steady, linear (additive) change — simple interest, evenly-spaced measurements, loop counters.

References

  1. BookSullivan, M. Algebra and Trigonometry, 10th ed. Ch. 12 — Sequences, Induction, and Probability.
  2. WebsiteWikipedia — Arithmetic progression