Arithmetic Sequences

​LT 8: I can identify and use the formulas for arithmetic and geometric sequences.

GOAL: To further understand the arithmetic sequence; both in using and writing the sequence.

Multiple Choice

What is the difference between f(n) and a_n?

• f(n) is a function (can be graphed) and a_n represents a sequence.

• Nothing, they are interchangeable.

• I need to find a few more values to determine the difference.

• We can find infinitely many values using a_n and only five or six for f(n).

Multiple Choice

Given f(n) = 3n - 7 is a sequence that can be defined as a_n = 3n - 7;

f(10) = 23 and

f(10 - 1) = f(9) = 20,

which of the following statements is true?

• f(9) is the a_n-1 for f(10)

• 20 is the a_n-1 for 23

• 23 is the a_n-1 for 20

• Why a_n?

​An Arithmetic Sequence

• ​The value of each term is the value of the PREVIOUS term with a constant added. (or subtracted)

• When we talk about terms we mean the VALUE at the term number.

  • ​Notation;

    • Term: a_n​

    • Previous term: a_n-1

​Consider f(n) = 3n - 7 again.

• ​1st term is the value of f(1)

• ​2nd term is the value of f(2)

• ​3rd term is the value of f(3)

• ​4th term is the value of f(4)

​SEQUENCE IS THEN:

{-4, -1, 2, 5, ...}

​

​YOU DON'T EVEN NEED THE FORMULA ANYMORE...

​So...

​The recursive definition of an arithmetic sequence.

• ​First - define what is getting added or subtracted to the previous term.

​3, 6, 9, 12, ...

​

• ​We are adding 3 to the previous term.

  • ​THIS IS WHAT WE CALL THE COMMON DIFFERENCE ("d")

Multiple Choice

Given the sequence;

1, 5, 9, 13, ...

which of the following is the recursive definition?

• $$a_n=a_{\left(n-1\right)}+4$$  

• $$a_n=a_{\left(n-1\right)}+4;\ a_1=1$$  

• $$a_n=n+4;\ a_1=1$$  

• $$a_n=1+4n;\ a_1=1$$  

Multiple Choice

Given the sequence;

6, 17, 28, 39, 50....

write the recursive definition.

• a_n=a_n-1+11
  a₁=6
• a_n=a_n-1-4
  a₁=6
• a_n=a_n-1+9
  a₁=6
• a_n=a_n-1+6
  a₁=6

​The explicit formula of an arithmetic sequence - YOU MAY WANT TO WRITE IT DOWN!

• ​Use the common difference, plug into the formula and simplify.

• ​3, 6, 9, 12, ...

• Common difference is still 3​

• ​a_n = 3 + (n - 1)(3)

• ​a_n = 3 + 3n - 3

• ​a_n = 3n (THE EXPLICIT FORMULA)

  • ​Now you can find any term value!

• ​All found by simplifying

Multiple Choice

Find the common difference: -34, -29, -24, -19, ...

• d = -5
• d = 5
• d = -6
• d = 6

Multiple Choice

Find the common difference: -34, -29, -24, -19, ...

Now, which of the following is the explicit formula of the sequence?

• a_n = 5 + (n-1)(-34)

• a_n = -34 + (n-1)(5)

• a_n = -34(n) + (n-1)(5)

• a_n = (n-1) + (-34)(5)

Multiple Choice

Given the sequence:  25, 21, 17, 13,... Write the explicit formula.  Then, SIMPLIFY your equation.

• a_n = 4n + 29
• a_n = -4n + 25
• a_n = -4n + 29
• a_n = 4n + 25

Multiple Choice

Given the sequence:

2, 8, 14, ...

Write the explicit formula.

(Hint:  Write your formula and then simplify it.)

• a_n= 2 + 6n

• a_n= -6 + 6n

• a_n= 6n - 4

• a_n= 4 - 6n

Multiple Choice

Let's use it...Given a₁ = 5 and d = 0.5, write the explicit formula and use it to find the 25th term

• 18
• 17
• 15
• 14

Multiple Choice

Find the 35th term in the arithmetic sequence: -10, -14, -18, -22, ...

• -216
• -148
• -146
• -150

​To Summarize

An arithmetic sequence is simply follows a linear pattern.

​

You need two parts to find the explicit formula of an arithmetic sequence; the first term and the common difference.

​

When one asks you to find the nth term of a sequence - you are looking for the EXPLICIT FORMULA.

​

​You can use the EXPLICIT FORMULA to find ANY term in the sequence!!!