Coordinate Algebra Arithmetic and Geometric

SequencesLearning Target:Students can use the explicit formula to find the nth term of a sequence.

Arithmetic Sequences

(Common difference)

Geometric Sequences(common ratio)

Explicit Formul

a

Recursive

FormulaExplicit Formula

Recursive

Formula

Each distance is 0.2 miles greater than the previous distance.

Example: The distance between you and a lightning strike can be approximated by using a sequence. Times

(s)Distance (mi)

1 0.22 0.43 0.64 0.85 1.06 1.27 1.48 1.6

*When the terms of a sequence differ by the same nonzero number d, the sequence is an arithmetic sequence and d is the common difference.

• A sequence is a list of numbers that may form a pattern. • Each number in a sequence is called a

term.

Arithmetic Sequence

A sequence of terms that have a common difference (d) between them-Subtract terms right to left

• 12, 8, 4, 0, …Step 1: Find the difference between successive terms.

• Step 2: Use the common difference to find the next 3 terms.

Determine whether each sequence appears to be an arithmetic sequence. If so, find the common difference and the next three terms in the sequence.

Add -4 to each term to find the next term. The sequence is arithmetic and the common difference is -4.

0-4 = -4, -4 – 4 = -8, -8 – 4 = -12

• 1, 4, 9, 16, …• Step 1: Find the difference

between successive terms.

• The difference between successive terms is not the same, so the sequence is not arithmetic.

1 + 3 = 4, 4 + 5 = 9, 9 + 7 = 16

Explicit Formula for Arithmetic Sequence

(Finding the nth term) 1 1na a n d

Explicit Formula for Arithmetic Sequences

a1 + (n-1)d = an (Allows us to find ANY term in a sequence)

a1= 1st term

n= the number of the term you are looking for

d= common difference

an= the value of the term that you are looking for

Geometric Sequence

A sequence of terms that have a common ratio (r) between them-Divide terms right to left

Explicit Formula

Formula used to find the nth term of a sequence

Explicit Formula for Geometric

Sequence 1

1n

na a r

Arithmetic or Geometric?

Example:-22, -15, -8, -1, …

Arithmeticd = 7

Arithmetic or Geometric?

Example:7, 4, 1, -2, -5

Arithmeticd = -3

Arithmetic or Geometric?

Example:256, 64, 16, 4, …

Geometricr = 1/4

Arithmetic or Geometric?

Example:

Geometricr = 2/3

4 , 83 ,169 ,3227 ,…

Find the common difference, the explicit formula, and the tenth

term.3, 9, 15, 21, …

1 1na a n d 3 1 6na n

d = 6

a10 = 57

𝑎10=3+(10−1 )6

Find the common ratio, the explicit formula, and the

seventh term.3, 1.5, 0.75, 0.375, …

11

nna a r

13 0.5 nna

7 17 3 0.5a

1.5 0.53r

7 0.0456875a

The fifth term is 1,792. The constant ratio is 4. Write the

explicit formula.5 1792 and 5 and 4a n r

17 4 nna

5 111792 4a

11

nna a r

17 a

HomeworkSequence

Practice WS