1. Ch. 8 – Sequences, Series, and Probability
8.1 – Sequences and Series

2. Sequences
Infinite sequence = a function whose domain is the set of positive integers
a1 , a2 , … , an are the terms of the sequence
Ex: Find the first four terms of an = 4 + 2n
a1 = 4 + 2(1) = 6
a2 = 8, a3 = 10, a4 = 12
You can enter this function into your calculator and use the TABLE to check your answers.

3. It may help to write out a table and look for patterns that way!
Ex: Write an expression to find the nth term of the sequence 1, 3, 5, 7, …
Hint: If the difference between two numbers is constant, the pattern is linear!
Since the difference is 2 each time, we get…
… an = 2n - 1
Ex: Write an expression to find the nth term of the sequence 3, 6, 11, 18, …
Hint: If the difference between two numbers changes by a constant rate, the pattern is quadratic!
an = n2 + 2
It may help to write out a table and look for patterns that way!

4. Recursive Function = a sequence that uses previous terms as inputs
Ex: If a0 = 1, a1 = 1, and an = an-2 + an-1 for n ≥ 2, find a5 .
a2 = = 2
a3 = = 3
a4 = = 5
a5 = = 8
Factorial Notation: n! = 1(2)(3)(4)…(n-1)(n)
Ex: Find the first 3 terms of an .
It’s the Fibonacci sequence!

5. Evaluate without a calculator.1 10 3 30 5

6. Evaluate. 1
n – 1 n2
n + 1 n

7. Summations
The sum of the first n terms of a sequence is represented by:
where k is the index (starting value) and n is the limit of the summation
Ex: Find
Add a1 through a4!

8. Properties of Sums:
If c is a constant, then…
The sum of the first n terms of a finite sequence is called a partial sum or a finite series
The sum of all terms in an infinite sequence is called an infinite series

9. Find the sum of the infinite series.
Hint: Write out a few terms in the sequence and add them up!
4/9
2/5 4 ∞