Math 4 Unit 1: Sequences and Series

Recursive Sequence

A sequence expressed in the form of an operation that is performed on the previous term of the sequence to find the next. To use a recursive sequence, the previous term must be given to calculate any following terms.

Example: Find the first 4 terms of the recursive sequence below.

Explicit Sequence

A sequence expressed in the form of an equation in terms of n, where n is the term's position in the sequence. This method of expressing a sequence does not require any previous terms to find following terms; it only requires you have the position of the term (n) you are looking for and the first term of the sequence.

Example: Find the first 4 terms of the explicit sequence below.

Arithmetic Sequence

Example: Find the 4th term of an Arithmetic sequence with a first term of 7 and a common difference of -4.

Arithmetic Series

Example: Find the sum of the first 10 terms of an Arithmetic sequence with a first term of 7 and a common difference of -4.

Geometric Sequence

Example: Find the 6th term of a Geometric sequence with a first term of 2 (a1) and a common ratio of 3 (r).

Geometric Series

Example: Find the 6th term of a Geometric series with a first term of 2 (a1) and a common ratio of 3 (r).

Application Problem

A cybersecurity technician has a starting salary $106,000. Per his contract with the company, he is given a 5% raise each year. If he works for 20 years, how much will he have earned total by the end?

a1 = 106,000 because the salary starts at $106,000

r = 105% or 1.05 because the salary increases by 5% each year

n = 20 because the technician is working 20 years

Infinite Sequences

Find the sum of the following sequence, if there is one:

1/20, 1/40, 1/80...

Convergent & Divergent Series

A convergent series has an r value with an absolute value less than 1, meaning the sequence will get closer and closer to 0 as it progresses. This means you can calculate the sum of the series to the infinite term, as there will be a finite sum.

For example, the geometric series in the previous question has an r value of 1/2 (|1/2|<1), because its terms are getting closer and closer to 0 (in this case, smaller and smaller)

On the other hand, a divergent series has an r value with an absolute value greater than 1, meaning the sequence will get farther and farther away from 0 as it progresses. Because of this, you cannot find the sum to an infinite term as the sequence has no limit to how large its terms it can get.

For example, this geometric series has an r value of 2 (|2|>1), which means the terms will get larger and larger, farther and farther away from 0, meaning there is no real way to find an infinite total of the series since there is no limit to how high it will go.