April 30 th copyright2009merrydavidson Happy Birthday to: 4/25 Lauren Cooper

9.1 Sequences & Series SEQUENCE: A list that is ordered so that it has a 1 st term, a 2 nd term, a 3 rd term and so on. example: 1, 5, 9, 13, 17, … a 1 = 1; a 2 = 5; a 3 = 9, etc. The nth term is denoted by: a n The domain of a sequence is the set of positive integers. The nth term is used to GENERALIZE about other terms.

The three dots mean that this sequence is INFINITE. example: 1, 5, 9, 13, 17, … example: 2, -9, 28, -65, 126 This is a FINITE sequence.

“Series” uses + signs. Arithmetic Sequence Arithmetic Series 3, 8, 13, 18,

Given a “rule” for a sequence, find the 1 st 5 terms. EXAMPLE 1:

Example 2: Write the first 4 terms of the sequence.

Example 3. Write the first six terms of the sequence if

Factorial Notation n! = n(n – 1)(n – 2)…1 Special case: 0! = 1 8 math/prb/4/enter = 40,320

Factorial Notation n! = n(n – 1)(n – 2)…1 Special case: 0! = 1

Summation Notation The Greek letter sigma, instructs you to add up the terms of the sequence. Example of sigma notation

Example 4. Starting with an i value of 1 and ending with an i value of 5, write the series, then add = 60 Find the sum of:

Example 5. Starting with an k value of 3 and ending with a k value of 6, write the expanded sum. Notice: k=3 to k =6 is 4 terms = 90 Find the sum of:

Example = 35 Find the sum of: Notice there are 5 terms here because you are starting at zero.

A sequence uses comma’s A series uses + signs Summation notation uses sigma sign

1, 5, 9, 13, 17, … The common difference is 4 When the difference between successive terms of a sequence is always the same number, the sequence is called arithmetic. In other words, the terms increase (or decrease) by adding a fixed quantity “d”.

Is this sequence arithmetic? Example 7: 2, -4, 8, -16, 32… No because we are multiplying by -2 each time.

Is this sequence arithmetic? Example 8: -5, 7, 19, 31,… yes because we are adding 12 each time.

Is the sequence defined by S n = 3n + 5 arithmetic? Example 9: Let n = 1, n = 2, n = 3, etc to generate the sequence. 8, 11, 14, 17… yes because we are adding 3 each time. Notice that the common difference is the “slope” of the function. Therefore linear functions are arithmetic!

Is the sequence defined by S n = 4 - n arithmetic? Example 10: Let n = 1, n = 2, n = 3, etc to generate the sequence. 3, 2, 1, 0, … yes because we are subtracting 1 each time. d = -1 a 1 = 3 Therefore linear functions are arithmetic!

Formula for the nth term of Arithmetic Sequence: “a” is the first term and “d” is the common difference

11) Write the nth term of the sequence 2, 7, 12, 17,….. Step 1: find the common difference 5 Step 2: write down the formula Step 3: fill in the formula with what you know

12) Write the nth term of the sequence -12, -9, -6, ….. Step 1: find the common difference 3 Step 2: write down the formula Step 3: fill in the formula with what you know

Use when you know the first term, number of terms and common difference Use when you know first term, last term, and number of terms Notice: Both formulas need first term and number of terms. SUMMATION FORMULAS:

13) Find the sum of the first 12 terms of: Find “d”.. d = 4 Pick which formula you want to use. Plug and Chug.

14) Find the indicated partial sum of: Find the 1 st term.. a 1 = -1 Pick which formula you want to use. Plug and Chug. Find the 2 nd term. a 2 = 1 Find “d”. d = 2 Find the last term. a 26 = 51