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Mom's Basement University

Arithmetic Sequences

What You’ll Learn

Section titled “What You’ll Learn”

In this lesson you’ll learn what an arithmetic sequence is, how to find the common difference, and how to write both the explicit and recursive formulas.

The Concept

Section titled “The Concept”

An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is always the same. This constant is called the common difference (d).

The diagram shows the sequence 7, 11, 15, 19, 23. Each green arc shows the common difference d = +4 being added to get the next term. Every arithmetic sequence works this way. You start with a first term and keep adding the same number.

Explicit formula (gives the nth term directly):

an=a1+(n−1)da_n = a_1 + (n - 1)d

Recursive formula (gives the next term from the previous one):

A recursive formula defines each term using the term before it. Instead of jumping straight to any term like the explicit formula does, you build the sequence one step at a time. To find the 5th term you need the 4th, to find the 4th you need the 3rd, and so on. That’s why a recursive formula always needs a starting value (a₁) to get going.

an=an−1+d(n≥2)a_n = a_{n-1} + d \quad (n \ge 2)

This says: “to get any term, take the previous term and add d.”

Worked Example

Section titled “Worked Example”

1. Find the 10th term of 7, 11, 15, 19, …

Here a₁ = 7 and d = 4.

a10=7+(10−1)(4)=7+36=43a_{10} = 7 + (10 - 1)(4) = 7 + 36 = 43

2. Write the explicit formula for 3, 7, 11, 15, …

a₁ = 3, d = 4.

an=3+(n−1)(4)=3+4n−4=4n−1a_n = 3 + (n - 1)(4) = 3 + 4n - 4 = 4n - 1

3. Given the recursive formula an = a(n−1) + 6 with a₁ = 2, find a₅

a2=2+6=8a3=8+6=14a4=14+6=20a5=20+6=26\begin{aligned} a_2 &= 2 + 6 = 8 \\[0.5em] a_3 &= 8 + 6 = 14 \\[0.5em] a_4 &= 14 + 6 = 20 \\[0.5em] a_5 &= 20 + 6 = 26 \end{aligned}

Real-World Application

Section titled “Real-World Application”

Arithmetic sequences model many everyday situations:

  • Saving a fixed amount each week or month
  • Salary with fixed annual raises
  • Monthly rent or bill payments that increase by a fixed amount
  • Seating arrangements in rows with constant increase per row
  • Simple linear depreciation of an asset

Example: you save 75 dollars every week. Your total savings after n weeks forms an arithmetic sequence with a₁ = 75 and d = 75.

Quiz

Section titled “Quiz”
What is the common difference in 4, 9, 14, 19, ...?
The explicit formula for an arithmetic sequence is:
Find the 8th term: $a_1 = 10$, $d = -3$
An arithmetic sequence has a constant:
Find the sum of the first 10 terms of the arithmetic sequence where $a_1 = 3$ and $d = 4$.