Ch. 8 – Sequences, Series, and Probability 8.1 – Sequences and Series
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.
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!
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 = 1 + 1 = 2 a3 = 1 + 2 = 3 a4 = 2 + 3 = 5 a5 = 3 + 5 = 8 Factorial Notation: n! = 1(2)(3)(4)…(n-1)(n) Ex: Find the first 3 terms of an . It’s the Fibonacci sequence!
Evaluate without a calculator. 1 10 3 30 5
Evaluate. 1 n – 1 n2 n + 1 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 a1 through a4!
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. Hint: Write out a few terms in the sequence and add them up! 4/9 2/5 4 ∞