Do Now

Identify something in life that uses it’s result as a new input.

Arithmetic Sequencing (Recursive Formula)

Recursive formula:

If A(n) is a current term, how can the term before it be represented?

If B(n + 1) is the current term, how can the term before it be represented?

Each problem is different. You must take into consideration the conditions set in the problem.

How can ‘the next term’ be represented?

Current term / Next term
an + 1
an

Identify if the formula is explicit or recursive. Then explain why.

Formula / Explicit / Recursive / Explain
A(n + 1) = A(n) + 5
A(n) = 2n + 3
A (n) = A(n + 1) – 7

In the explicit formula (from yesterday), the first term of the sequence can be found by substituting the n with 1 and n 1is declared. What is the first term for the following arithmetic sequence?

an = 5 + 2nwhen n 1

The recursive formula does not reveal the initial term, so it must be declared with the equation. You must also state n 1 … ALWAYS!

an + 1 = an – 8 when a1 = 14 and n 1

Example: Find the first five terms in the sequence f (n + 1) = -2f (n) + 1, where f (1) = 8 andn 1.

Example: If B(n + 1) = 33 and B(n) = 28, write a possible recursive formula involving B(n + 1) and B(n) that would generate 28 and 33 in the sequence.

What does B(17) – B(16) mean?

Example: Write the recursive formula for f(n) = -3n + 2 for n 1.

Example: Write the explicit and the recursive formula for

Exercise:

1. / List the first five terms of the sequence from A(n + 1) = A(n) – 19 for n 1 and A(1) = -6
2. / Write a recursive formula for: Doug accepts a job where his starting salary will be $30,000 per year, and each year he will receive a raise of $3,000.
3. / Consider a sequence given by the formula an = an – 1 – 5, where a1= 12 and n 1.
  1. What kind of formula is given: explicit or recursive?
  1. List the first five terms of the sequence.
  1. Write an explicit formula.
  1. Find a6 and a100 of the sequence.

4. / 9, 1, -7, -15, …
  1. Write an explicit formula for the sequence.
  1. Write a recursive formula for the sequence.
  1. Find the 38th term.

Name: ______Date: ______

Algebra I CCExit Ticket

1. List the first five terms defined by the sequence for an + 1 = an + 6, where a1 = 11 for n 1.

2. Write a recursive formula for the sequence 14, 21, 28, 35, …

Name: ______Date: ______

Algebra I CCExit Ticket

1. List the first five terms defined by the sequence for an + 1 = an + 6, where a1 = 11 for n 1.

2. Write a recursive formula for the sequence 14, 21, 28, 35, …

Name: ______Date: ______

Algebra I CCHW #29

Lesson Summary
The Arithmetic Explicit Formula is good for finding a term far away from a sequence:
an = a1 + (n – 1)d
Recursion is good for finding a term close to the sequence. Be flexible as to referencing the prior term:
If an is the current term, the prior term is an – 1.
If an + 1 is the current term, the prior term is an.
1. / List the first five terms in A(n + 1) = A(n) + 4 for n 1 and A(1) = -2.
2. / Write a recursive formula for 8, 9, 10, 11, 12, …
3. / Write a recursive formula for -101, -91, -81, -71, …
4. / Using the explicit formula an = 2n + 10,
  1. Generate the first 5 terms of the sequence.
  1. Write a recursive formula for the sequence.

Review:

5. / Simplify: 3 + 2(5 – 1)2