Notes 9.4 – Sequences and Series

I. Sequences A.) A progression of numbers in a pattern. 1.) FINITE – A set number of terms 2.) INFINITE – Continues forever

B.) Notation - a k where a is the sequence and k refers to the k th term. C.) Explicitly defined sequence– Allows us to substitute k into an equation to find the k th term. D.) Recursively defined sequence– Defines each term by using the previous term.

E.) Ex. 1 - Define the following sequence both explicitly and recursively.

II. Types of Sequences A.) Arithmetic Sequence – any sequence with a common difference between terms. B.) Geometric Sequence - any sequence with a common ratio between terms.

C.) Ex. 2- Define the following sequence both explicitly and recursively.

D.) Constructing a Sequence – Ex. 3 - The third and sixth terms of a sequence are -12 and 48 respectively. Find an explicit formula for the sequence if it is 1.) arithmetic and 2.) geometric 1.) -

2.) Divide the two equations

III. Fibonacci Sequence A.) Summation of successive terms. Can only be defined recursively B.) Ex. 4– a k = 1, 1, 2, 3, 5, 8, 13,…

IV. Summation Notation A.) The way we express the sum of a sequence of n terms {a 1, a 2, a 3, …, a n }  “the sum of a k from k = 1 to n” k = the index of summation.

B.) Ex. 5- C.) Ex. 6-

V. Sum of a Finite Arith. Seq. A.)

B.) Proof -

C.) Ex. 7- Find the sum of the following sequences: A.) 3, 6, 9, 12,…, 21B.) 111, 108, 105,…27

VI. Sum of a Finite Geom. Seq. A.)

B.) Ex. 8- Find the sum of the following sequence: 3, 6, 12, 24,…,48, 96 C.) Ex. 9- Find the sum of the following sequence: 10, 20, 30, 40,…

D.) Ex. 9 - Find the sum of the following sequence: 0.1, 0.01, 0.001, , …

VII. Sum of a Infinite Geom. Seq. A.) A.K.A. an INFINITE SERIES B.) Not a true sum. This is considered a Partial Sum If it converges…

If | r |< 1 C.) Ex. 10- Find the sum of the following series: