Sequences & Series

-1 + (n-1)-3

-1, -4, -7, -10, . . . y = _______

-1 + (n-1)-4

-1, -5, -9, -13, . . . y = _______

Find the next three terms in the sequence: 4, -1, -6, ...

-11, -16, -21

-12 + (n-1)3

-12, -9, -6, -3, . . . y = _______

-2 + (n-1)-2

-2, -4, -6, -8, . . . y = _______

-2 + (n-1)-3

-2, -5, -8, -11, . . . y = _______

Find the 12th term in a sequence with common difference -8 and first term 58.

-30

-6 + (n-1)4

-6, -2, 2, 6, . . . y = _______

Find the common difference in the sequence: 10, 3, -4, -11, ...

-7

Initial population of 600 penguins triples in size each year for six years. Find size of population after six years

145,800

Find the first term in a sequence where the common difference is 6 and the eleventh term is 77.

17

Find the next three terms in the sequence: 3, 7, 11, 15, ...

19, 23, 27

Find sum to infinity of the geometric series; 3- ³/₂ + ³/₄ -...

2

Find the 10th term in the sequence: -5, -2, 1, 4, ...

22

Find sum to infinity of the geometric series; 5 + 1+ 1/5+...

25/4

Find the 72nd term in the sequence: -5, -1, 3, 7, ..

279

3 + (n-1)-3

3, 0, -3, -6, . . . y = _______

3 + (n-1)-2

3, 1, -1, -3, . . . y = _______

Geometric Sequence

5,25,125,525, ..... is an example of a(an)

Find sum of the first eight terms of the geometric series; ½+¼+¹/₈ +...

510/512

First Five Terms: −38, −138, −238, −338, −438

Given the explicit formula for an arithmetic sequence find the first 5 terms.

First Five Terms: −4, 3, 10, 17, 24

Given the explicit formula for an arithmetic sequence find the first 5 terms.

First Five Terms: −9.2, −11.3, −13.4, −15.5, −17.6

Given the explicit formula for an arithmetic sequence find the first 5 terms.

Convergent

If the ratio of an infinite geometric series is geater than 0 and less than 1 then the series is

Divergent

If the ratio of an infinite geometric seriesgreater than 1 the series is

Is the pattern an arithmetic sequence? Why? 2, 4, 8, 16, _______, _______

No, each number is the previous number multiplied by 2

Is the pattern an arithmetic sequence? Why? 2, -4, 8, -16, _______, _______

No, each term is the previous number multiplied by -2.

Is the pattern an arithmetic sequence? Why? 3, 9, 27, 81, _______, _______

No, each term is the previous number multiplied by 3

Is the pattern an arithmetic sequence? Why? 1, 4, 9, 16, _______, _______

No, each term is the square of the next consecutive integer, starting with 1.

Explicit Formula for Arithmetic Sequence

Relates any term in a sequence to only the first term and the common difference.

Explicit Formula for Geometric Sequence

Relates any term in a sequence to only the first term and the common ratio.

sigma

Symbol for sum

200

The first term of an arithmetic sequence is 2 and the common difference is 4. Find the sum of the first 10 terms

term number

The order number in which the term falls in the sequence (x)

28

The sum of a sequence with 7th term -2 and 12th term 8 is 364. Find the number of terms in the sequence

1/3 or .333...

The sum of the geometric series: 0.3 + 0.03 + 0.003 + . . .

Series

The sum of the terms of a sequence is called a(an)

2048

What is the 18th term of the series 1/64,1/32,1/16,1/8,1/4.....

1/4

What is the common ratio? 64, 16, 4 , 1...

A(n) = 5 + (n − 1)(8)

Write a recursive formula for the sequence: 5, 13, 21, 29...

A(n) = 51 + (n − 1)(−4)

Write a recursive formula for the sequence: 51,47,43,39...

Is the pattern an arithmetic sequence? Why? 2, 5, 8, 11, _______, _______

Yes, each number is the previous number plus 3

Is the pattern an arithmetic sequence? Why? 9, 15, 21, 27, _______, _______

Yes, each term is the previous number plus 6

arithmetic sequence

a numerical pattern that adds or subtracts at a constant rate or value

a_1

a sub 1 or the first term in sequence a

a_n

a sub n or the nth term in sequence a

term

a value in a sequence

Find the explicit form: 5, 2, -1, -4, -7, ....

a_n= 5 + (n-1)-3

Arithmetic Sequence

adding or subtracting the same number

sequence

an ordered list of values

has a common difference

arithmetic sequence

Term

each number in the sequence

45/2

find the sum of the geometric series 30, -10, 10/3, -10/9, ...

Explicit

plug in term number

common difference

the difference between any two consecutive terms in an arithmetic sequence

Recursive

uses previous term, when defining using this process you must define at least one previous term

The Geometric Sequence Infinite Sum

where a is the first term and |r| < 1

The first two terms of a geometric series are 216+54. Find common ration

¼

Find sum of the first eight terms of the geometric series; 3+6+12+...

765

Find the common difference: 16, 25, 34, ..

9

Find sum of first 30 odd natural numbers ( 1+3+...+59)

900

Find the 21st term in the sequence. -4, 1, 6, 11, 16, ...

96

-3

An arithmetic sequence has 15 terms. The first term is 30 and the last term is -12. Find the common difference

915

An arithmetic sequence has first term 12, and the third term is 26. Find the sum of the first 15 terms.

220

Archie uses an exercise bike each time he goes to the gym. The first time he goes he uses the bike for 10 minutes. Each time he uses the bike he increases his time by 2 minutes. He aims to eventually use the bike for 30 minutes. How many minutes will he have spent on the bike in total when he reaches 30 minutes?

Common difference

Arithmetic sequences always have a

4100

Calculate 5 + 10 + ... + 195 + 200

Geometric sequences always have a

Common Ratio

Geometric

Determine if the sequence is arithmetic, geometric or neither: 100, 50, 25, 12.5

1/5

Determine the common ration: 500, 100, 20

Common Difference d

Determines Arithmetic sequence

Common Ratio r

Determines Geometric sequence

125

Find f(1) for the sequence: 125, 25, 5...

54

Find f(3) for the given sequence: 6, 18, 54...

486

Find f(6) for the given sequence: 2, 6, 18...

-7

Find the common difference: -7,-14,-21,-28...

11

Find the common difference: 12,23,34,45...

(1/5)

Find the common ratio of the sequence: 1/4, 1/20, 1/100

3

Find the common ratio of the sequence: 4, 12, 36

(1/3)

Find the common ratio of the sequence: 81, 27, 9

4

Find the common ratio: 1, 4, 16, 64

-2.

Find the common ration of -2/5, 4-5, -8/5, ..

t(n) = 48(1/2)ⁿ⁻¹

Find the equation of the sequence: 48, 24, 12, 6, 3

f(n)=1(4)ⁿ⁻¹

Find the explicit rule for the sequence: 1, 4, 16...

f(n)=125(1/5)ⁿ⁻¹

Find the explicit rule for the sequence: 125, 25, 5...

1, -4, 16, -64, 256

Find the first five terms of the sequence: t(n) = 1(-4)ⁿ⁻¹

243, 729

Find the next 2 numbers 3, 9, 27, 81

-192, 768, -3072

Find the next three terms of the sequence: 3, -12, 48

f(1)=3; f(x)=f(x-1)×2

Find the recursive equation for the geometric sequence: 3, 6, 12, 24

-1275

Find the sum of 4+1+(-2)+...+-89

610

Find the sum of the first 10 terms of the sequence 2, 5, 8, 11

-2364

Find the sum of the first 24 terms of the arithmetic progression, if the first term is 5 and the common difference is -9

38.130.

Find the sum of the first seven term of the series 32/27+16/9+8/3+...

-29

Find u₂₀ for the sequence 9, 7, 5, 3, ...

First Five Terms: 15/8, 19/8, 23/8, 27/8, 31/8

Given the explicit formula for an arithmetic sequence find the first 5 terms.

First Five Terms: 28, 38, 48, 58, 68

Given the explicit formula for an arithmetic sequence find the first 5 terms.

First Five Terms: 35, 39, 43, 47, 51

Given the explicit formula for an arithmetic sequence find the first 5 terms.

First Five Terms: −34, −44, −54, −64, −74

Given the explicit formula for an arithmetic sequence find the first 5 terms.

First Five Terms: −35, −135, −235, −335, −435

Given the explicit formula for an arithmetic sequence find the first 5 terms.