Recursive and Explicit Forms of Arithmetic Sequences

1. Arithmetic Sequences

2. Section 1.3 Arithmetic Sequences
An arithmetic sequence is a sequence in which
the difference between each term and the
preceding term is always constant.
Which of the following sequences is arithmetic?
a. {14, 10, 6, 2, -2, -6, -10, . . . }
b. {3, 5, 8, 12, 17, . . . }
a. yes, the difference between each term is -4
b. no, the difference between the first two terms is 2
and the difference between the 2nd and 3rd term is 3.

3. Recursive Form of an Arithmetic
Sequence
Un = Un-1 + d
for some constant d and all n > 2
The number d is called the common difference of the
arithmetic sequence.

4. Graph of an Arithmetic Sequence
• If {Un} is an arithmetic sequence with U1 = 3
and U2 = 4.5 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.

5. •
a. Find the common difference.
• U - U = 4.5 - 3 = 1.5
2 1
• The common difference is 1.5

6. • b. Write the sequence as a recursive function.
•U 1 = 3, Un = Un-1 + 1.5, for n > 2
First Method for Always one
Term finding nth term greater than
by using the subscript of first
preceding term. term.

7. • c. Give the first six terms of the
sequence.

8. Explicit Form of an Arithmetic Sequence
• In an arithmetic sequence {Un} with common
difference d, Un = U1 + (n - 1)d for every n > 1
Find the nth term of an arithmetic sequence with first term -5 and
common difference of 3. Sketch a graph of the sequence.
Un = U1 + (n - 1)d
= -5 + (n - 1)3
= -5 + 3n - 3
= 3n - 8

9. Find the nth term of an arithmetic sequence with first term -5 and
common difference of 3. Sketch a graph of the sequence.

10. 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?
First find d; d = 9 - 5 = 4
Second find explicit form: Un = 5 + (n - 1)4
Un = 4n + 1
Then find 45th term: U45 = 4(45) + 1
U45 = 181

11. 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.
To find d when given to non-consecutive terms use the formula:
d = Um --Un
m n
d = 93 -- 57
10 6
=9

12. Finding U1
Select either of the given terms and substitute
into Explicit formula. Un = U1 + (n - 1)d
U6 = 57 U10 = 93
57 = U1 + (6 - 1)9 93 = U1 + (10 - 1)9
U1 = 12 U1 = 12

13. FORMULAS
Explicit Form Recursive Form
Un = U1 + (n - 1)d Un = Un-1 + d
Un = 12 + (n - 1)9 Un = Un-1 + 9, for n > 2
Un = 9n + 3, for n > 1