This document explains arithmetic sequences, characterized by a repeated addition of a constant called the common difference. It outlines how to express these sequences using both explicit and recursive formulas, providing examples to illustrate the calculations involved. Additionally, it introduces the concept of series as the sum of sequence terms and discusses the computation of partial sums.

1. Arithmetic Sequences
Sequence is a list of numbers typically with a
pattern.
2, 4, 6, 8, …
The first term in a sequence is denoted as a1,
the second term is a2, and so on up to the nth
term an.
Each number in the list called a term.
a1, a2, a3, a4, …

2. Finite Sequence has a fixed number of
terms. {2, 4, 6, 8}
A sequence that has infinitely many
terms is called an infinite sequence.
{2, 4, 6, 8,…}
Algebraically, a sequence can be written as an
explicit formula or as a recursive formula.
Explicit formulas show how to find a specific term
number (n).
Recursive formula show how to get from a given term (an-
1) to the next term (an)

3. An Arithmetic Sequence is a sequence where
you use repeated addition (with same number)
to get from one term to the next.
Ex: 4, 1, -2, -5, … is an arithmetic sequence
-3 -3 -3
The number that needs to be added each time to get to the next
term is called the common difference
The common difference for the above arithmetic sequence is -3 .

4. Explicit formula for
Arithmetic Sequence:
an = + (n - 1)d
Recursive formula for
an Arithmetic
Sequence:
a1 = #
an = an-1 + d
Common
difference
Explicit Formula
Substitute the values:
an = 4 + (n – 1)(- 3)
So the explicit formula
is: an = -3n + 7
The Recursive Formula
is:
a1 = 4
an = an-1 – 3
For the example: 4, 1, -2, -5, …
First term

5. A series is the sum of ALL the terms of a
sequence. (can be finite or infinite)
A partial sum is the sum of the first n terms of
a series…denoted Sn
𝑆𝑛=
𝑛 ( 𝑎1 +𝑎𝑛)
2
Number of terms
First term Last term
How do you add these
sequences of numbers?

6. For the example: 4, 1, -2, -5, …
1) Find S4.
2) Find S20. (Think….)

7. Example: For the arithmetic sequence 2, 6, 10, 14, 18, …
a) Write the explicit formula for the sequence.
b) Write the recursive formula for the sequence.
c) Find the 15th
partial sum of the sequence (S15).
an =
a1 = #
an = an-1 + d
𝑆𝑛=
𝑛(𝑎1 +𝑎𝑛)
2