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): $a<em>n = a</em>1 + (n-1)d$ where $a_1$ is the first term and $n\in\mathbb{N}$.
• Recursive form: $a<em>n = a</em>{n-1} + d$ 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, ⋯
  • $a<em>1 = 3,\ d = 5\,;\ a</em>n = 3 + (n-1)\cdot 5$
  • Next term: $a_6 = 3 + 5\cdot 5 = 28$
• Sequence 4, 7, 10, 13, 16, ⋯
  • $a<em>1 = 4,\ d = 3\; a</em>n = 4 + (n-1)\cdot 3$
  • Next term: $a_6 = 4 + 5\cdot 3 = 19$

Translate Between Recursive and Explicit Forms

• From recursive to explicit: $a<em>n = a</em>1 + (n-1)d$
• From explicit to recursive: $a<em>n = a</em>{n-1} + d,\ a<em>1 = a</em>1$
• Example: if $a<em>2 = a</em>1 + 0.5$, then $a<em>n = a</em>1 + 0.5(n-1)$

Solve Problems With Arithmetic Sequences

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

Sigma Notation

• Sum of a sequence can be written as $\sum<em>{i=1}^{n} a</em>i$ where $a_i$ is the i-th term.
• Replace $a_i$ with the explicit formula when using sigma notation.
• Example: series $2 + 9 + 16 + \dots + 79$
  • $a<em>1 = 2,\ d = 7,\ a</em>n = 79,\ n = 12$
  • 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)
  • $a<em>1 = 1,\ a</em>{10} = 10,\ d = 1$
  • Sum: $S_{10} = \frac{10}{2}\,(1 + 10) = 55$
• Use explicit formula when you know $a<em>1, a</em>n,\text{ or }d$

Concept Summary

• Arithmetic sequences: $a<em>n = a</em>1 + (n-1)d$
• Common difference: $d$
• Recursive vs Explicit representations
• Arithmetic series (sums): $S<em>n = \frac{n}{2}\,(a</em>1 + a<em>n) = \frac{n}{2}\,(2a</em>1 + (n-1)d)$
• Sigma notation: $\sum<em>{i=1}^{n} a</em>i$

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.