Arithmetic-Sequences-and-Series-Sums.pptx

sequence and series (arithmetic)

1. Arithmetic Sequences Sequence is a list of numbers typically with a pattern. 2, 4, 6, 8, … The first term in a sequence is denoted as a 1 , the second term is a 2 , and so on up to the nth term a n . Each number in the list called a term . a 1 , a 2 , a 3 , a 4 , …

2. Finite Sequence has a fixed number of terms. {2, 4, 6, 8} A sequence that has infinitely many terms is called an infinite sequence . {2, 4, 6, 8,…} Algebraically, a sequence can be written as an explicit formula or as a recursive formula . Explicit formulas show how to find a specific term number (n). Recursive formula show how to get from a given term (a n-1 ) to the next term (a n )

3. An Arithmetic Sequence is a sequence where you use repeated addition (with same number) to get from one term to the next. Ex: 4, 1, -2, -5, … is an arithmetic sequence -3 -3 -3 The number that needs to be added each time to get to the next term is called the common difference The common difference for the above arithmetic sequence is -3 .

4. Explicit formula for Arithmetic Sequence: a n = + (n - 1)d   Recursive formula for an Arithmetic Sequence: a 1 = # a n = a n-1 + d Common difference Explicit Formula Substitute the values: a n = 4 + (n – 1)(- 3) So the explicit formula is: a n = -3n + 7 The Recursive Formula is: a 1 = 4 a n = a n-1 – 3 For the example: 4, 1, -2, -5, … First term

5. A series is the sum of ALL the terms of a sequence. (can be finite or infinite) A partial sum is the sum of the first n terms of a series…denoted S n   Number of terms First term Last term How do you add these sequences of numbers?

6. For the example: 4, 1, -2, -5, … 1) Find S 4 . Find S 20 . (Think….)

7. Example: For the arithmetic sequence 2, 6, 10, 14, 18, … Write the explicit formula for the sequence. Write the recursive formula for the sequence. c) Find the 15 th partial sum of the sequence (S 15 ). a n =   a 1 = # a n = a n-1 + d