8.1 – Sequences and Series

Sequences Infinite sequence = a function whose domain is the set of positive integers a 1, a 2, …, a n are the terms of the sequence Ex: Find the first four terms of a n = 4 + 2n a 1 = 4 + 2(1) = 6 a 2 = 8, a 3 = 10, a 4 = 12 You can enter this function into your calculator and use the TABLE to check your answers.

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… … a n = 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! a n = n It may help to write out a table and look for patterns that way!

Recursive Function = a sequence that uses previous terms as inputs Ex: If a 0 = 1, a 1 = 1, and a n = a n-2 + a n-1 for n ≥ 2, find a 5. a 2 = = 2 a 3 = = 3 a 4 = = 5 a 5 = = 8 Factorial Notation: n! = 1(2)(3)(4)…(n-1)(n) Ex: Find the first 3 terms of a n. It’s the Fibonacci sequence!

Evaluate without a calculator

Evaluate n – 1 3. n 2 4. n n

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 a 1 through a 4 !

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

Find the sum of the infinite series /9 3. 2/ ∞ Hint: Write out a few terms in the sequence and add them up!