Sequences

​

​

​

​

​

​

​

​

​

​

​

​

​

​How many triangles would there be in the 12th term?

Sequences Review

By Christen Calhoun

• have a common ratio

• you multiply or divide each term by the same number to get the next term

• think exponential function​

​Geometric sequence

• have a common difference

• ​you add or subtract the same number to get the next term

• think linear function​

​Arithmetic sequence

Arithmetic vs geometric

​Notice the pattern between the y values.​ When looking at arithmetic sequences, the y values represent the sequence numbers, and the x values represent the positions of the numbers.

​Arithmetic sequences are like lines with the y = mx+b formula that you are all familiar with

​a = the difference between consecutive terms

n = ​the term position we are looking at

​a + b always equals the first term

​

​Any questions? Alright, now it's your turn...

​Every arithmetic sequence has an nth term rule in

the form of an+b

(AKA the sequence formula)​

​

​ It would look like this in linear form

y = 2x+6​

Multiple Choice

What is the sequence formula of 8, 6, 4, 2?

• -2

• 2n-10

• -2n+10

• -2n

Multiple Choice

How do you write this sequence in linear form?

10, 15, 20, 25

• y = 5x

• y = 5x+5

• y = 5x+1

• y = x+5

• Allows you to find the value of a specific term based on it's position​

• ​a_{n =} a₁ + d(n-1)

​explicit

• Allows you to find the value of a specific term based on the previous term

• ​a_{n =} a_n-1 +d

​Recursive

​

two sequence formulas

​

a_n = nth term, unknown n = term position

a₁ = 1st term d = common difference ​

Lets look at a few examples

Find ​the 15th term in the sequence 4, 8, 12, 12, 16, 20...

​

a_n = a₁​ + d(n-1)

a₁₅ = 4 + 4(14)

a_{15 =} 60​

Is everyone good with this? Get ready, it's your turn again to answer a few questions​...

​Which explicit equation represents this sequence? -23, -25, -27, -28... (a_n = a₁+d(n-1))

​

a. a_n = -28+2(n-1)

b. a_n = -23+2(n-1)

c. a_n = -23-2(n-1)

​

​

​

​

Multiple Choice

What is the 100th term in this sequence?

11, 19, 27, 35, 43...

• 803

• 1,111

• 1,097

• 819

Multiple Choice

Which equation represents this sequence in recursive formula (a_n = a_n-1 + d)?

__, __, 11, 14, __

• a_n = a₁+3(4)

• a_n = a₄-14

• a_n = a_n-1+3

Multiple Choice

Which explicit formula represents this sequence ?

3, 12, 21, 30, 39, __

• a_n = a₅+9

• a_n = 3+9(n-1)

• a_n = 3+9(n+1)

​Can be written as exponential functions

​

Geometric sequences

Any geometric sequence can be written in exponential form

​y = a₁r^x-1

a₁ = first term (same as arithmetic)

r = common ratio ​

Any questions on this? Your turn!​

​Example

3, 12, 48, 192

​y = 3(4^x-1)​

Multiple Choice

Write this geometric sequence in exponential form (y = a₁r^x-1)

1, 6, 36, 216...

• y = 1(6^x-1)

• y = 6(6^x-1)

• y = 6(1^x)

Multiple Choice

Which sequence represents this equation?

y = 2(3^x-1)

• 2, 4, 8, 16...

• 6, 18, 54, 162...

• 2, 6, 18, 54...

recursive and explicit form for geometric sequences

Recursive

a_n = ​a_n-1(r)

Ex.

5, 10, 20, 40...

a_n = a_n-1(2)​

Did you get these written down on your notes?​ It's your turn to try it out!

​Explicit

a_n = ​a₁(r^n-1)

Ex.

5, 10, 20, 40...

a_n = 5(2^n-1)​

Multiple Choice

Which equation represents the sequence in recursive form?

120, 60, 30, 15...

• a_n = a₁(2^n-1)

• a_n = a_n-1(1/2)

• a_n = a_n-1(2)

Multiple Choice

Which equation represents the sequence in explicit form?

2, 8, 32, 128...

• a_n = 4(2^n-1)

• a_n = a_n-1(4)

• a_n = 2(4^n-1)

Multiple Choice

Find the 14th term of the sequence 3, 9, 27, 81,...

• 14,348,907

• 4,782,969

• 1,594,323

• 243

Multiple Choice

Find the first four terms of the sequence given the rule:

a_n = 4(5)^n-1

• 4, 20, 100, 500

• 5, 20, 80, 320

• 20, 100, 500, 2500

• 4, 9, 14, 19

Multiple Select

Select all that represent this sequence

2, 18, 162, 1458

• y = 2(9^x-1)

• a_n = 2(9^n-1)

• a_n = a_n-1(2)

• a_n = a_n-1(9)

Multiple Choice

Is the following sequence arithmetic or geometric?

-3, -7, -11, -15

• arithmetic

• geometric

• neither

• both

Multiple Choice

Is the following sequence arithmetic or geometric?

5, 15, 45, 135,...

• arithmetic

• geometric

• neither

• both

Poll

On a scale of 1 to 5, 5 being extremely confident, how confident do you feel in your ability to ace a test on sequences?

• 1

• 2

• 3

• 4