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​

• 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)

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​Can be written as exponential functions

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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)​

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)​