Arithmetic & Geometric Sequences – Section 14.2
ARITHMETIC SEQUENCES
An ________________ ________________ is a sequence in which each term differs from the proceeding term
by a constant amount ____. The constant ____ is called the ______________ __________________ of the
sequence.
Ex. 2, 6, 10, 14, 18,…
first term: 𝑎1 =
common difference, d =
Example 1: Write the first five terms of the arithmetic sequence whose first term is 7 and whose common
difference is 2.
****The general term 𝑎𝑛 of an arithmetic sequence is given by
where 𝑎1 is the first term and d is the common difference.
Example 2: Consider the arithmetic sequence whose first term is 3 and common difference is −5.
a. Write an expression for the general term 𝑎𝑛 .
b. Find the twentieth term of this sequence.
OYO: Consider the arithmetic sequence whose first term is 2 and whose common difference is −3.
a. Write an expression for the general term 𝑎𝑛 .
b. Find the twelfth term of the sequence.
Example 3: Find the eleventh term of the arithmetic sequence whose first three terms are 2, 9, and 16.
OYO: Find the ninth term of the arithmetic sequence whose first three terms are 3, 9, and 15.
Example 4: If the third term of an arithmetic sequence is 12 and the eighth term is 27, find the fifth term.
OYO: If the third term of an arithmetic sequence is 23 and the eighth term is 63, find the sixth term.
Example 5: Donna had an offer for a job starting at $40,000 per year and guaranteeing her a raise of $1600 a
year for the next 5 years. Write the general term for the arithmetic sequence that models Donna’s potential
annual salaries, and find her salary for the fourth year.
GEOMETRIC SEQUENCES
A __________________ ________________ is a sequence in which each term is obtained by multiplying the
preceding term by a constant ____. The constant ____ is called the ____________ __________ of the
sequence.
Ex. 12, 6, 3, 3/2, …
first term: 𝑎1 =
common ratio, r =
Example 6: Write the first five terms of a geometric sequence whose first term is 7 and whose common ratio is
2.
The general term 𝑎𝑛 of a geometric sequence is given by
where 𝑎1 is the first term and r is the common ratio.
Example 7: Find the eighth term of the geometric sequence whose first term is 12 and whose common ratio is
1
.
2
1
OYO: Find the seventh term of the geometric sequence whose first term is 64 and whose common ratio is 4
Example 8: Find the fifth term of the geometric sequence whose first three terms are 2, -6, and 18.
OYO: Find the fifth term of the geometric sequence whose first three terms are -3, 6, and -12.
5
5
Example 9: If the second term of a geometric sequence is 4 and the third term is 16, find the first term and the
common ratio.
9
OYO: If the second term of a geometric sequence is 2 and the third term is
common ratio.
27
4
, find the first term and the
Example 10: The population size of a bacterial culture growing under controlled conditions is doubling each
day. Predict how large the culture will be at the beginning of day 7 if it measures 10 units at the beginning of
the day 1.