Arithmetic and Geometric Sequences
Name____________________________ 11-1
A sequence is a list of numbers in a particular order. Each number in a sequence is called a term. The
first term is symbolized by a1 , the second term is symbolized by a 2 , and so on. There are two major
types of explicit sequences, arithmetic and geometric.
1. Arithmetic Sequences
An arithmetic sequence is a sequence in which each term after the first is found by adding a
constant, called the common difference, d, to the previous term. The terms between any two
non-consecutive numbers of an arithmetic sequence are called arithmetic means.
Arithmetic
Formula for the nth term:
an  a1  d (n  1) where
a1 is the initial term and
d is the common difference
Equation Type
Linear
Let’s look at the arithmetic sequences from the “Do Now”. Identify the first term and the common
difference, then use them to write an explicit equation.
first term ( a1 )
common
difference (d)
explicit equation
a) 4, 7, 10, _____, 16, . . .
________
_______
___________________
b) 3.5, 6, 8.5, 11, _____, . . .
________
_______
___________________
c) _____, -5.5, -9.5, -13.5, . . .
________
_______
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d) Given an equation, you can find the sequence:
an  5n  1
Write the first 7 terms of the sequence: ___________________________________________________
e) Given
a1  10
and the common difference of the arithmetic sequence is 6.
Write the first 4 terms of the sequence: __________________________________________
Write the explicit equation of this sequence: ______________________________________

In the arithmetic sequences and equations, where do you see the common difference?
_______________________________________________________________________
Summary
Example #1: Find the next term
Find the next 4 terms of the arithmetic sequence 55, 49, 43, …
Find the common difference d by subtracting two consecutive terms. d = _______
Now add that to the third term of the sequence and continue adding.
The next 4 terms are: _______, _______, _______, _______
Example #2: Find a particular term
Find the 10th term in the following sequence: { 2, 6, 10, 14, 18, …}
Find the common difference: d = _______
a1 = _______ (first in sequence)
Substitute known values in the equation and solve:
an  a1  d (n  1)
Example #3: Write an equation for the nth term
Write an equation for the nth term of the arithmetic sequence 8, 17, 26, 35, …
In this sequence, a1 = _______ and d = _______.
Use the nth term formula to write an equation. Then simplify.
an  a1  d (n  1)
2. Geometric Sequences
Geometric – a sequence of numbers such that each number after the first is found by
multiplying the previous one by a constant called the common ratio, r. The terms between any
two non-consecutive numbers of an geometric sequence are called geometric means.
Equation Type –
Geometric
Formula for the nth term:
a n  a1 (r ) n 1 where
a1 is the initial term and
r is the common ratio
Exponential
Let’s look at the geometric sequences from the “Do Now”. Identify the first term and the common ratio,
then use them to write an explicit equation.
first term ( a1 )
common
explicit equation
ratio (r)
a) 2, 4, 8, 16, _____, . . .
________
_______
___________________
b) 1, _____, 9, 27, 81, . . .
________
_______
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c) 8, 12, 18, _____, 40.5, . . .
________
_______
___________________
d) 256, 64, 16, 4, _____, . . .
________
_______
___________________
e) -4, 8, -16, 32, -64, _____, . . .
________
_______
___________________

In the geometric sequences and equations, where do you see the common ratio?
_______________________________________________________________________
f) Given an equation, you can find the sequence
an  4(2) n 1
Write the first 6 terms of the sequence: ___________________________________________________
g) Given a1
 10 and the common ratio of this geometric sequence is 6.
Write the first 3 terms of the sequence:
____________________________________________
Write the explicit equation of this sequence:
____________________________________________
Summary
Example #1: Find the next term
Find the next term in the geometric 8, 20, 50, 125, …
Find the common ratio r by dividing two consecutive terms. r = _______
Now multiply the last term of the sequence by that ratio.
The next term is: _______
Example #2: Find a particular term
Find the 10th term in the following sequence: { 2, 6, 18, 54, 162, …}
Find the common ratio: r = _______
a1 = _______ (first in sequence)
Substitute known values in the equation and solve:
a n  a1 (r ) n 1
Example #3: Write an equation for the nth term
Write an equation for the nth term of the geometric sequence 3, 12, 48, 192, …
In this sequence, a1 = _______ and r = _______.
Use the nth term formula to write an equation.
a n  a1 (r ) n 1
Example #4: Find a term given another term and the ratio
Find the 10th term of a geometric sequence for which a 4 = 108 and r = 3.
First, find the value of a1 .
Now find a10 .
Arithmetic and Geometric Means
Arithmetic Means – Sometimes you are given two terms of a sequence, but they are not consecutive
terms. The terms between any two non-consecutive terms of an arithmetic sequence are called
arithmetic means.
Example A: Find the 4 arithmetic means between 16 and 91.
First, use the nth term formula to find the common difference.
Now, use the value of d to find the four arithmetic means.
Geometric Means – the missing terms between two non-consecutive terms of a geometric sequence
are call geometric means.
Example B: Find 3 geometric means between 2.25 and 576.
First, use the nth term formula to find the common ratio.
Now, use the value of r to find the four geometric means. (Since there are two possible
common ratios, there are two possible sets of geometric means.)