SEQUENCES & SERIES

Explicit & Recursive Formulas

ARITHMETIC & GEOMETRIC SEQUENCES A Sequence is a list of things

(usually numbers) that are in order.

2 Types of formulas: Explicit & Recursive Formulas

ARITHMETIC SEQUENCES

A list of numbers that related to each other by a rule.

The terms are the numbers that form the sequence.

Goes from one term to the next by adding and subtracting.

Two types of sequences with two types of formulas.

1. Arithmetic: Explicit & Recursive

2. Geometric: Explicit & Recursive

ARITHMETIC SEQUENCES A sequence in which the common

difference between terms is always the same number.

ADD to get to the next term.

2 types of formula to find either the NEXT number or A NUMBER in the sequence.

Explicit vs Recursive

RECURSIVE FORM Finds the NEXT Term in the sequence

an = an-1 + d

Previous TermCommon Difference

• the nth term (term you want)• n = the term number• means the number before nth term • d = common difference (can be

negative)

EXAMPLES – FINDING THE COMMON DIFFERENCE (D) Find the common difference.

1) 3, 5, 7, 9, 112) -2, -4, -6, -8, -103) -3, 0, 3, 6, 94) 19, 10, 1, -8, -17

RECURSIVE FORM IS…

NEXT Number = CURRENT Term + d

an = an-1 + d

RECURSIVE FORM EXAMPLES1) Find the 4th term given:

a4 = ? , a3 = 6 , d = 2

2) Find the 3rd term given:a3 = ? , a2 = 17 , d = -7

3) Find the 6th term given:a6 = ? , a5 = 2 , d = -5

WRITE THE RECURSIVE FORM FOR EACH SEQUENCE. THEN FIND THE NEXT THREE TERMS.

4) 12, 9, 6, 3, 0, …

5) 56, 61, 66, 71, …

6) -14, -24, -34, -44, …

EXPLICIT FORMULA Finds any number in the sequence.

the nth term (term you want) n = the term number

d = common difference (can be negative) )

First TermCommon Difference

One less term than the term number

EXAMPLE #1Find the 10th term of our sequence: 4, 6, 8, 10…

1. Identify all your terms. n = _____ = _____ d = _____

2. Substitute values into our formula a a a 22

EXAMPLE #2Find the 25th term of our sequence: 6, 10, 14, 18, …

1. Identify all your terms.n = _____ a = _____ d = _____

2. Substitute values into our formula aaa

YOU TRY Consider the sequence {6, 17, 28, 39,

50, …} Find the 12th term.

Find the 50th term.

Find the 100th term.

APPLICATION A bag of dog food weighs 8 pounds at

the beginning of day 1. Each day, the dogs are feed 0.1 pound of food. How much does the bag of dog food weigh at the beginning of day 30?

STEPS1. Determine if the sequence is Arithmetic-Do you add or subtract by the same amount from one term to the next?

2. Ask if you want to find the next term or a term in the sequence?

3. Find the common difference.-The number you add or subtract

4. Create a Recursive or Explicit formula-State first term or previous term,

GEOMETRIC SEQUENCES A sequence in which the ratio between

consecutive terms is always the same number.

A geometric sequence is formed by multiplying a term in the sequence by a fixed number to find the next term.

Common Ratio- the fixed number that you multiply each term in the sequence.

GEOMETRIC SEQUENCES A sequence in which the common ratio

between terms is always the same number.

MULTIPLY to get to the next term.

2 types of formula to find either the NEXT number or A NUMBER in the sequence.

Explicit vs Recursive

RECURSIVE FORM Finds the NEXT Term in the sequence

an = an-1 ∙ r

Previous TermCommon Ratio

• the nth term (term you want)• n = the term number• means the number before nth term • r = common ratio (can be a fraction)

FIND THE COMMON RATIO OF EACH SEQUENCE.

1. 750, 150, 30, 6,…

2. 2. 9, -36, 144, -576,…

3. 8, -24, 72, -216,…

WRITE THE RECURSIVE FOR EACH PROBLEM.

1. 750, 150, 30, 6,…

2. -3, -6, -12, -24,…

3. 9, -36, 144, -576,…

EXPLICIT FORMULA Finds any number in the sequence.

the nth term (term you want) n = the term number

r = common ratio (can be a fraction)

First TermCommon Ratio

One less term than the term number

FIND A10, A33 , A5

1. 750, 150, 30, 6,…

2. -3, -6, -12, -24,…

3. 9, -36, 144, -576,…

4. 8, -24, 72, -216,…

STEPS1. Determine if the sequence is Geometric-Do you multiply or divide by the same amount from one term to the next?

2. Ask if you want to find the next term or a term in the sequence?

3. Find the common ratio.-The number you multiply or divide

4. Create a Recursive or Explicit formula-State first term or previous term,

GRAPHS Arithmetic Graphs: LINEAR

Geometric Graphs: EXPONENTIAL

ARITHMETIC MEAN

Of any two number is the average of the two numbers.

Use the arithmetic mean to find a missing term of an arithmetic sequence.

Two terms of an arithmetic sequence and their arithmetic mean lie on the same line.

GEOMETRIC SEQUENCE Of any two positive numbers by taking the

positive square root of the product of the two numbers.

Use the geometric mean to find a missing term of a geometric sequence.

ARITHMETIC & GEOMETRIC SERIES

A indicated Sum of terms of a sequence is called a Series

OR…. A Sum of an infinite sequence it is

called a "Series" (it sounds like another name for

sequence, but it is actually a sum).

EXAMPLEArithmetic Sequence

Arithmetic Series

3, 6, 9, 12, 15 3 + 6 + 9 + 12 +1 5

-4, -1, 2 -4 + (-1) + 2

ARITHMETIC SERIES

Sum of Numbers in Sequence

Must Know first term, last term, and number of terms

Number of terms

First term

• is the first term

• n is the number of terms

nth term

EXAMPLE 1: Find the sum of the first 50 positive

integers.

EXAMPLES:

1. Write the related series for each finite sequence then evaluate.21, 18, 15, 12, 9, 6, 3

Solution:

2. Evaluate the series: (8 terms)5, 13, 21…61

Remember: These are the SUM of the numbers.

GEOMETRIC SERIES

is the first term r is the "common ratio" between terms n is the number of terms

Need to know three things:  the first term, how many terms to add and the common ratio!!

EXAMPLE 2: Find the sum of the first six terms of a

geometric series for which

EXAMPLES:1. Evaluate series to the given term.

2. Evaluate series.…, n = 6

*Don’t forget we are adding the terms not multiply by .5 each term.

EXAMPLE 3: