Module 3 Lesson 2: Recursive Formulas For Sequences

Consider Akelia’s sequence 5, 8, 11, 14, 17, ….

What is the formula for this sequence? Label what each part of the formula does.

Akelia writes the following:

5

8=5+3

11 = 5 +3 +3 =5 + 2x3

14 = 5 + 3 + 3 + 3 = 5x3x3x3

Let’s rewrite the sequence from above…

5

8 = 5 + 3

11 = 8 + 3

14 = 11 + 3

17 = 14 + 3

Johnny looks at this and writes

A(n + 1) = A(n) + 3. What do

you think this means?

What do we call the 5th term?

What do we call the 5th term

in terms of the 4th term?

How could we find the (n+1)th

term in terms of the nth

term?

Heads up! This is not A*(n+1), it is not

something that can be distributed.

Akelia, in a playful mood, asked Johnny: “What would happen if we change the ‘+’

sign in your formula to a ‘−’ sign? To a ‘×’ sign? To a ‘÷’ sign?”

Ben made up a recursive formula and used it to generate a sequence. He used

𝐵(𝑛) to stand for the 𝑛th term of his recursive sequence.

What does 𝐵(3) mean?

What does 𝐵(𝑚) mean?

Why does Akelia’s formula

have a times three in it, but

Johnny’s have a +3?

If we wanted to know what

the 200th term in the

sequence was, whose formula

would we use?

If we wanted to know how the

sequence changes from term-

to-term, whose formula

would we use?

What sequence does

𝐴(𝑛 +1) = 𝐴(𝑛) −3 for 𝑛 ≥

1 and 𝐴(1) = 5 generate?

What sequence does 𝐴(𝑛

+1) = 𝐴(𝑛) ⋅3 for 𝑛 ≥ 1

and 𝐴(1) = 5 generate?

What sequence does

𝐴(𝑛+1) = 𝐴(𝑛)÷3 for 𝑛 ≥ 1

and 𝐴(1) = 5 generate?

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.

Consider a sequence given by the formula 𝑎𝑛 = 𝑎(𝑛−1) −5, where 𝑎1 = 12 and 𝑛 ≥

2. a. List the first five terms of the sequence.

b. Write an explicit formula.

c. Find 𝑎6 and 𝑎100 of the sequence.

One of the most famous sequences is the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13,

21, 34, …. 𝑓(𝑛 +1) = 𝑓(𝑛)+𝑓(𝑛 −1), where 𝑓(1) = 1, 𝑓(2) = 1, and 𝑛 ≥ 2.

How is each term of the sequence generated?

Each sequence below gives an explicit formula. Write the first five terms of each

sequence. Then, write a recursive formula for the sequence.

𝑎𝑛 = 2𝑛 +10 for 𝑛 ≥ 1

For each sequence, write either an explicit or a recursive formula.

1, −1, 1, −1, 1, −1, …

Module 2 Lesson 3: Arithmetic and Geometric Sequences

Identify the sequence as arithmetic or geometric, and write a recursive formula for the sequence. Be sure to identify

your starting value.

14, 21, 28, 35, … 49, 7, 1, 1

7 ,

1

49, …

4, 40, 400, 4000, … 8. −101, −91, −81, −71, …

ARITHMETIC SEQUENCE: A sequence is

called arithmetic if there is a real number 𝑑

such that each term in the sequence is the

sum of the previous term and 𝑑.

GEOMETRIC SEQUENCE: A sequence is

called geometric if there is a real number 𝑟

such that each term in the sequence is a

product of the previous term and 𝑟.