1. Chapter 1: Number Patterns 1.3: Arithmetic Sequences
Essential Question: What is the symbol for summation?

2. 1.3 Arithmetic Sequences
Arithmetic Sequence → a sequence where the difference between each term and the preceding term is constant (adding/subtracting the same number repeatedly)
Example 1: Arithmetic Sequence
Are the following sequence arithmetic?
If so, what is the common difference?
{ 14, 10, 6, 2, -2, -6, -10 }
{ 3, 5, 8, 12, 17 }
Yes; common difference is -4
Not arithmetic

3. 1.3 Arithmetic Sequences
In an arithmetic sequence {un} (Recursive Form): un = un-1 + d for some constant d and all n ≥ 2
Example 2: Graph of an Arithmetic Sequence
If {un} is an arithmetic sequence with u1=3 and u2=4.5 as the first two terms
Find the common difference
Write the recursive function
Give the first seven terms of the sequence
Graph the sequence

4. 1.3 Arithmetic Sequences Example 2: Graph of an Arithmetic Sequence
If {un} is an arithmetic sequence with u1=3 and u2=4.5 as the first two terms
Find the common difference
Write the recursive function
Give the first seven terms of the sequence
Graph the sequence
d = 4.5 – 3 = 1.5
u1 = 3 and un = un for n ≥ 2
3, 4.5, 6, 7.5, 9, 10.5, 12
See page 22, figure 1.3-1b. Note that the graph is a straight line (this will become relevant later)

5. 1.3 Arithmetic Sequences Explicit Form of an Arithmetic Sequence
Arithmetic sequences can also be expressed in a form in which a term of the sequence can be found based on its position in the sequence
Example 3: Explicit Form of an Arithmetic Sequence
Confirm that the sequence un = un with u1=-7 can also be expressed as un = -7 + (n-1) ∙ 4

6. 1.3 Arithmetic Sequences Example 3, continued
Explicit Form of an Arithmetic Sequence
un = u1 + (n-1)d for every n ≥ 1
u1 → 0 “4”s
u2 → 1 “4”
u3 → 2 “4”s
u4 → 3 “4”s
u5 → 4 “4”s

7. 1.3 Arithmetic Sequences Example 4 Solution
Find the nth term of an arithmetic sequence with first term -5 and common difference 3
Solution
Because u1 = -5 and d = 3, the formula would be: un = u1 + (n-1)d un = = n – =
So what would be the 8th term in this sequence? The 14th? The 483rd?
-5 + (n-1)3
3n - 8

8. 1.3 Arithmetic Sequences Formula: un = u1 + (n – 1)(d)
Example 5: Finding a Term of an Arithmetic Sequence
What is the 45th term of the arithmetic sequence whose first three terms are 5, 9, and 13?
Solution
u1 = 5
The common difference, d, is 9-5 = 4
u45 = u1 + (45 – 1)d = 5 + (44)(4) = 181

9. 1.3 Arithmetic Sequences
Example 6: Finding Explicit and Recursive Formulas
If {un} is an arithmetic sequence with u6=57 and u10=93, find u1, a recursive formula, and an explicit formula for un
Solution:
The sequence can be written as: …57, ---, ---, ---, 93, ...
u6, u7, u8, u9, u10
The common difference, d, can be found like the slope

10. 1.3 Arithmetic Sequences
The value of u1 can be found using the explicit form of an arithmetic sequence, working backwards: un = u1 + (n-1)d u6 = u1 + (6-1)(9) We don’t know u1 , but we know u = u – 45 = u = u1
Now that we know u1 and d, the recursive form is given by: u1 = 12 and un = un-1 + 9, for n ≥ 2
The explicit form of the arithmetic sequence is given by: un = 12 + (n-1)9 un = n – 9 un = 9n + 3, for n ≥ 1

11. 1.3 Arithmetic Sequences
Today’s assignment: Page 29, 1-6, Ignore directions for graphing
Tomorrow: Summation Notation Calculators will be important

12. Chapter 1: Number Patterns 1.3: Arithmetic Sequences (Day 2)
Essential Question: What is the symbol for summation?

13. 1.3 Arithmetic Sequences Summation Notation
When we want to find the sum of terms in a sequence, we use the Greek letter sigma: ∑
Example 7

14. 1.3 Arithmetic Sequences Computing sums with a formula
Either of the following formulas will work:
What they mean:
Add the first & last values of the sequence, multiply by the number of times the sequence is run, divide by 2
Number of times sequence is run (k) multiplied by 1st value. Add that to k(k-1), divided by 2, multiplied by the common difference

15. Version A
Using those formulas:

16. Version B

17. 1.3 Arithmetic Sequences Computing sums with the TI-86
Storing regularly used functions:
2nd, Catlg-Vars (Custom), F1, F3
Move down to function desired, store with F1-F5
We will need sum & seq( for this section
Using sequence:
seq(function, variable, start, end)
Using sum:
sum sequence

18. 1.3 Arithmetic Sequences
seq (function, variable, start, end) sum seq(7-3x,x,1,50) = -3475

19. 1.3 Arithmetic Sequences Partial Sums
The sum of the first k terms of a sequence is called the kth partial sum
Example 9: Find the 12th partial sum of the arithmetic sequence: -8, -3, 2, 7, …
Formula method:

20. 1.3 Arithmetic Sequences
Example 9: Find the 12th partial sum of the arithmetic sequence: -8, -3, 2, 7, …
Calculator method:
Need to determine equation: un=-8+(n-1)(5) [you can simplify] -8+5n-5=5n-13
sum seq(5x-13,x,1,12)

21. Assignment Page: 29 – 30 Problems: 7 – 17 & 31 – 43 (odd)
Write the problem down
Make sure to show either
The formula you used when solving the sum; or
What you put into the calculator to find the answer