1. Introduction to Arithmetic Sequences
18 May 2011

2. Arithmetic Sequences
When the difference between any two numbers is the same constant value
This difference is called d or the constant difference
{4, 5, 7, 10, 14, 19, …}
{7, 11, 15, 19, 23, ...}
← Not a Sequence
← Arithmetic Sequence
d = 4

3. Your Turn:
Determine if the following sequences are arithmetic sequences. If so, find d (the constant difference).
{14, 10, 6, 2, –2, …}
{3, 5, 8, 12, 17, …}
{33, 27, 21, 16, 11,…}
{4, 10, 16, 22, 28, …}

4. Recursive Form un = un–1 + d n ≥ 2
The recursive form of a sequence tell you the relationship between any two sequential (in order) terms.
un = un–1 + d n ≥ 2
common difference

5. Writing Arithmetic Sequences in Recursive Form
If given a term and d
Substitute d into the recursive formula

6. Examples: Write the recursive form and find the next 3 terms
u1 = 39, d = 5

7. Your Turn: Write the recursive form and find the next 3 terms
u1 = 8, d = –2
u1 = –9.2, d = 0.9

8. Writing Arithmetic Sequences in Recursive Form, cont.
If given two, non-sequential terms
1. Solve for d
d = difference in the value of the terms
difference in the number of terms
2. Substitute d into the recursive formula

9. Example #1
Find the recursive formula
u3 = 13 and u7 = 37

10. Example #2
Find the recursive formula
u2 = –5 and u7 = 30

11. Example #3
Find the recursive formula
u4 = –43 and u6 = –61

12. Your Turn Find the recursive formula:
1. u3 = 53 and u5 = u2 = -7 and u5 = 8
3. u3 = 1 and u7 = -43

13. Explicit Form un = u1 + (n – 1)d n ≥ 1
The explicit form of a sequence tell you the relationship between the 1st term and any other term.
un = u1 + (n – 1)d n ≥ 1
common difference

14. Summary: Recursive Form vs. Explicit Form
un = un–1 + d n ≥ 2
Sequential Terms
Explicit Form
un = u1 + (n – 1)d n ≥ 1
1st Term and Any Other Term

15. Writing Arithmetic Sequences in Explicit Form
You need to know u1 and d!!!
Substitute the values into the explicit formula
1. u1 = 5 and d = u1 = -4 and d = 5

16. Writing Arithmetic Sequences in Explicit Form, cont.
You may need to solve for u1 and/or d.
Solve for d if necessary
Back solve for u1 using the explicit formula
u4 = 12 and d = 2

17. Example #2
u7 = -8 and d = 3

18. Example #3
u6 = 57 and u10 = 93

19. Example #4
u2 = -37 and u7 = -22

20. Your Turn: Find the explicit formulas:
1. u5 = -2 and d = u11 = 118 and d = 13
3. u3 = 17 and u8 = u2 = 77 and u5 = -34

21. Using Explicit Form to Find Terms
Just substitute values into the formula!
u1 = 5, d = 2, find u5

22. Using Explicit Form to Find Terms, cont.
u1 = -4, d = 5, find u10

23. Your Turn: 1. u1 = 4, d = ¼ 2. u1 = -6, d = ⅔ Find u8 Find u4