College Algebra

Arithmetic and Geometric Sequences

Lesson

A sequence is an ordered list of numbers. Two especially common kinds appear everywhere in algebra:

Arithmetic sequences add the same constant to get from one term to the next. That constant is the common difference $d$ . Example: $3,\,7,\,11,\,15,\,\ldots$ with $d = 4$ .
Geometric sequences multiply by the same constant each step. That constant is the common ratio $r$ . Example: $2,\,6,\,18,\,54,\,\ldots$ with $r = 3$ .

The two formulas:

\text{Arithmetic: } a_n = a_1 + (n - 1)\,d

\text{Geometric: } a_n = a_1 \cdot r^{\,n - 1}

$a_1$ is the first term, and $a_n$ is the $n$ -th term. Notice the $(n-1)$ rather than $n$ — there are $n - 1$ steps between the first term and the $n$ -th.

Worked example 1 — arithmetic

a_1 = 2,\ d = 5,\ a_{10} = ?

a_{10} = 2 + (10 - 1)(5) = 2 + 45 = 47

Worked example 2 — geometric

a_1 = 3,\ r = 2,\ a_5 = ?

a_5 = 3 \cdot 2^{5 - 1} = 3 \cdot 16 = 48

If you’re given two non-consecutive terms instead of $a_1$ and the difference (or ratio): set up two equations using the formula and solve. For example, if $a_3 = 11$ and $a_7 = 23$ , subtract to find $4d = 12$ , so $d = 3$ .

How to type your answer

Type a single number — the term value, the common difference, or the count, depending on what the question asks. Use a slash for fractions. Examples: 47, 1/32, 3, 16.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

3,\,7,\,11,\,15,\,\ldots\ \ a_5 = ?

Problem 2

2,\,6,\,18,\,54,\,\ldots\ \ a_5 = ?

Problem 3

5,\,8,\,11,\,14,\,\ldots\ \ a_6 = ?

Problem 4

1,\,2,\,4,\,8,\,\ldots\ \ a_5 = ?

Practice

Standard problems matching the lesson.

Problem 5

\text{Arithmetic, } a_1 = 2,\ d = 5,\ a_{10} = ?

Problem 6

\text{Geometric, } a_1 = 3,\ r = 2,\ a_5 = ?

Problem 7

\text{Arithmetic, } a_1 = 10,\ d = -3,\ a_8 = ?

Problem 8

\text{Geometric, } a_1 = 64,\ r = \tfrac{1}{2},\ a_5 = ?

Problem 9

4,\,7,\,10,\,13,\,\ldots\ \ a_{20} = ?

Problem 10

3,\,6,\,12,\,24,\,\ldots\ \ a_6 = ?

Problem 11

\text{Arithmetic, } a_1 = 1,\ d = 4,\ a_{15} = ?

Problem 12

\text{Geometric, } a_1 = 2,\ r = 3,\ a_4 = ?

Problem 13

100,\,90,\,80,\,\ldots\ \ a_{12} = ?

Problem 14

1,\,\tfrac{1}{2},\,\tfrac{1}{4},\,\tfrac{1}{8},\,\ldots\ \ a_6 = ?

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 15

\text{Arithmetic, } a_3 = 11,\ a_7 = 23.\ d = ?

Problem 16

\text{Arithmetic, } a_5 = 14,\ a_9 = 26.\ a_1 = ?

Problem 17

\text{Geometric (}r > 0\text{), } a_2 = 6,\ a_4 = 54.\ r = ?

Problem 18

\text{Geometric, } a_3 = 12,\ a_6 = 96.\ r = ?

Problem 19

\text{Arithmetic, } a_1 = -3,\ d = 4,\ a_{25} = ?

Problem 20

\text{Geometric, } a_1 = 5,\ r = 2,\ a_8 = ?

Problem 21

\text{Geometric, } a_1 = 729,\ r = \tfrac{1}{3},\ a_4 = ?

Problem 22

5,\,8,\,11,\,\ldots,\,50.\ \text{How many terms?}