Arithmetic Sequences and Series — Quick Reference

Understand Arithmetic Sequences

• An arithmetic sequence has a constant difference between consecutive terms, called the common difference d.
• General term (explicit form): an = a1 + (n-1)d where a_1 is the first term and n\in\mathbb{N}.
• Recursive form: an = a{n-1} + d with initial term a_1.
• Distinction: explicit gives any term directly; recursive uses the previous term.

Write a Function for an Arithmetic Sequence

• Recursive function form: f(1) = a_1\,,\quad f(n) = f(n-1) + d\,.
• Explicit function form: f(n) = a_1 + (n-1)d\,.
• Example structure: a term is defined by the previous term plus d (with an initial condition).

Examples (conceptual)

• Sequence 3, 8, 13, 18, 23, ⋯
  • a1 = 3,\ d = 5\,;\ an = 3 + (n-1)\cdot 5
  • Next term: a_6 = 3 + 5\cdot 5 = 28
• Sequence 4, 7, 10, 13, 16, ⋯
  • a1 = 4,\ d = 3\; an = 4 + (n-1)\cdot 3
  • Next term: a_6 = 4 + 5\cdot 3 = 19

Translate Between Recursive and Explicit Forms

• From recursive to explicit: an = a1 + (n-1)d
• From explicit to recursive: an = a{n-1} + d,\ a1 = a1
• Example: if a2 = a1 + 0.5, then an = a1 + 0.5(n-1)

Solve Problems With Arithmetic Sequences

• Example: Auditorium seating
  • Given a1 = 18, a5 = 26; difference per row: d = \frac{a5 - a1}{5-1} = \frac{26-18}{4} = 2
  • Explicit: a_n = 18 + 2(n-1)
  • Then a_{12} = 18 + 2\cdot 11 = 40
• Sum of arithmetic sequence
  • Finite sum: Sn = \frac{n}{2}\,(a1 + an) or Sn = \frac{n}{2}\,(2a_1 + (n-1)d)
• Example: Sum of 1,4,7,10,13
  • a1 = 1,\ d = 3,\ a5 = 13,\ n = 5
  • S_5 = \frac{5}{2}\,(1 + 13) = 35

Sigma Notation

• Sum of a sequence can be written as \sum{i=1}^{n} ai where a_i is the i-th term.
• Replace a_i with the explicit formula when using sigma notation.
• Example: series 2 + 9 + 16 + \dots + 79
  • a1 = 2,\ d = 7,\ an = 79,\ n = 12
  • Explicit term: a_i = 2 + 7(i-1) = 7i - 5
  • Sigma form: \sum_{i=1}^{12} (7i - 5) = 486

Use a Finite Arithmetic Series

• Pyramid example: 1, 2, 3, …, 10 rows (top row 1, bottom row 10)
  • a1 = 1,\ a{10} = 10,\ d = 1
  • Sum: S_{10} = \frac{10}{2}\,(1 + 10) = 55
• Use explicit formula when you know a1, an,\text{ or }d

Concept Summary

• Arithmetic sequences: an = a1 + (n-1)d
• Common difference: d
• Recursive vs Explicit representations
• Arithmetic series (sums): Sn = \frac{n}{2}\,(a1 + an) = \frac{n}{2}\,(2a1 + (n-1)d)
• Sigma notation: \sum{i=1}^{n} ai

Do You Understand?

• Essential Question: What is an arithmetic sequence, and how do you represent and find its terms and their sums?
• Vocabulary: Distinguish between arithmetic sequence (terms) and arithmetic series (sum).
• Error Analysis: A claim like 0, 1, 3, 6, 10 is not arithmetic; the differences are 1, 2, 3, 4, so the pattern is not constant.
• Precision: How to calculate terms and sums accurately using the explicit or recursive forms.