Arithmetic Sequences Lesson 1.3

Arithmetic Sequence This is a sequence in which the difference between each term and the preceding term is always constant. {10, 7, 4, 1, -2, -5,…} Is {2, 4, 7, 11, 16,…} an arithmetic sequence? Recursive Form of arithmetic sequence u n = u n-1 + d For some constant d and all n ≥ 2

Example If {u n } is an arithmetic sequence with u 1 = 2.5 and u 2 = 6 as its first two terms a. Find the common difference b. Write the sequence as a recursive function c. Give the first six terms of the sequence d. Graph the sequence

Explicit Form of Arithmetic Sequence In an arithmetic sequence {u n } with common difference d, u n = u 1 + (n-1)d for every n ≥ 1. If u 1 = -5 and d = 3 we can find the explicit form by, u n = -5 + (n-1)3 = n – 3… leaving us with the explicit form of 3n - 8

Example If we wanted to know the 38 th term of the arithmetic sequence whose first three terms are 15, 10, and 5, how would we do that?

Here’s How u n = u 1 + (n-1)d = 15 + (38-1)(-5) = 15 + (-5)(37) = = -170 Lets look at example 6 on page 24 because it is far too exhaustive to write down!

Summation Notation What is the sum of this sequence?

Graphing Calculator Exploration We are going to use the sum sequence key on our graphing calculators Find the sum of this little diddy

Partial Sums of Arithmetic Sequences If {u n } is an arithmetic sequence with common difference d, then for each positive integer k, the kth partial sum can be found by using either of the following formulas There is a proof on this on page 27…if anybody really cares

Example Find the 14 th partial sum of the arithmetic sequence 21, 15, 9, 3,… U 14 = u 1 + (14 – 1)(-6) = 21 + (13)(-6) = 21 + (-78) = -57

Find the Sum of all multiples of 4 from 4 to 404! We know that we are adding …, so 4x1, 4x2, 4x3, … and we can take 404 ÷ 4 to get the 101 term. What this means is there is 101 multiples of 4 in between 4 and 404 u 1 =4, k=101, and u 101 = 404! Use form 1

Here is a little story about Larry Larry owns an automobile dealership. He spends $18,000 on advertising during the first year, and he plans to increase his advertising expenditures by $1400 in each subsequent year. How much will Larry spend on advertising during the first 9 years?

Now…Get To Work Slackers!