# Algebra 2

## Educators

DF
AG
sw

### Problem 1

Describe each pattern formed. Find the next three terms.
$$80,77,74,71,68, \dots$$

DF
Daisy F.

### Problem 2

Describe each pattern formed. Find the next three terms.
$$4,8,16,32,64, \dots$$

Susmith B.

### Problem 3

Describe each pattern formed. Find the next three terms.
$$0,3,7,12,18, \dots$$

AG
Ankit G.

### Problem 4

Describe each pattern formed. Find the next three terms.
$$1,4,7,10,13, \dots$$

Susmith B.

### Problem 5

Describe each pattern formed. Find the next three terms.
$$100,10,1,0.1,0.01, \dots$$

sw
Sarah W.

### Problem 6

Describe each pattern formed. Find the next three terms.
$$\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \frac{1}{32}, \dots$$

Susmith B.

### Problem 7

Describe each pattern formed. Find the next three terms.
$$4,-8,16,-32,64, \ldots$$

DF
Daisy F.

### Problem 8

Describe each pattern formed. Find the next three terms.
$$1,2,6,24,120, \dots$$

Susmith B.

### Problem 9

Describe each pattern formed. Find the next three terms.
$$0,1,0, \frac{1}{3}, 0, \frac{1}{5}, \dots$$

DF
Daisy F.

### Problem 10

Fractal Geometry Draw the first four figures of the sequence described.
_____ is replaced by

AG
Ankit G.

### Problem 11

Fractal Geometry Draw the first four figures of the sequence described.
(TRIANGLE NOT COPY) is replaced by (TRIANGLE NOT COPY)

AG
Ankit G.

### Problem 12

Write a recursive formula for each sequence. Then find the next term.
$$-2,-1,0,1,2, \ldots$$

Susmith B.

### Problem 13

Write a recursive formula for each sequence. Then find the next term.
$$43,41,39,37,35, \ldots$$

DF
Daisy F.

### Problem 14

Write a recursive formula for each sequence. Then find the next term.
$$40,20,10,5, \frac{5}{2}, \dots$$

Susmith B.

### Problem 15

Write a recursive formula for each sequence. Then find the next term.
$$6,1,-4,-9, \dots$$

DF
Daisy F.

### Problem 16

Write a recursive formula for each sequence. Then find the next term.
$$144,36,9, \frac{9}{4}, \dots$$

Susmith B.

### Problem 17

Write a recursive formula for each sequence. Then find the next term.
$$\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \frac{1}{32}, \dots$$

DF
Daisy F.

### Problem 18

Write an explicit formula for each sequence. Then find $a_{12}$
$$4,5,6,7,8, \dots$$

Susmith B.

### Problem 19

Write an explicit formula for each sequence. Then find $a_{12}$
$$\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \frac{1}{6}, \dots$$

DF
Daisy F.

### Problem 20

Write an explicit formula for each sequence. Then find $a_{12}$
$$4,7,10,13,16, \dots$$

Susmith B.

### Problem 21

Write an explicit formula for each sequence. Then find $a_{12}$
$$3,7,11,15,19, \ldots$$

DF
Daisy F.

### Problem 22

Write an explicit formula for each sequence. Then find $a_{12}$
$$-2 \frac{1}{2},-2,-1 \frac{1}{2},-1, \ldots$$

Susmith B.

### Problem 23

Write an explicit formula for each sequence. Then find $a_{12}$
$$2,5,10,17,26, \dots$$

DF
Daisy F.

### Problem 24

Decide whether each formula is explicit or recursive. Then find the first five terms of each sequence.
$$a_{n}=2 a_{n-1}+3, \text { where } a_{1}=3$$

Susmith B.

### Problem 25

Decide whether each formula is explicit or recursive. Then find the first five terms of each sequence.
$$a_{n}=\frac{1}{2}(n)(n-1)$$

DF
Daisy F.

### Problem 26

Decide whether each formula is explicit or recursive. Then find the first five terms of each sequence.
$$(n-5)(n+5)=a_{n}$$

Susmith B.

### Problem 27

Decide whether each formula is explicit or recursive. Then find the first five terms of each sequence.
$$a_{n}=-3 a_{n-1}, \text { where } a_{1}=-2$$

DF
Daisy F.

### Problem 28

Decide whether each formula is explicit or recursive. Then find the first five terms of each sequence.
$$a_{n}=-4 n^{2}-2$$

Susmith B.

### Problem 29

Decide whether each formula is explicit or recursive. Then find the first five terms of each sequence.
$$a_{n}=2 n^{2}+1$$

DF
Daisy F.

### Problem 30

Decide whether each formula is explicit or recursive. Then find the first five terms of each sequence.
$$a_{n}=5 n$$

Susmith B.

### Problem 31

Decide whether each formula is explicit or recursive. Then find the first five terms of each sequence.
$$a_{n}=a_{n-1}-17, \text { where } a_{1}=340$$

DF
Daisy F.

### Problem 32

Entertainment Suppose you are building a tower of cards with levels as displayed below. Complete the
table, assuming the pattern continues.

Susmith B.

### Problem 33

Find the next two terms in each sequence. Write a formula for the $n$ th term. Identify each formula as explicit or recursive.
$$5,8,11,14,17, \dots$$

DF
Daisy F.

### Problem 34

Find the next two terms in each sequence. Write a formula for the $n$ th term. Identify each formula as explicit or recursive.
$$3,6,12,24,48, \dots$$

Susmith B.

### Problem 35

Find the next two terms in each sequence. Write a formula for the $n$ th term. Identify each formula as explicit or recursive.
$$1,8,27,64,125, \dots$$

DF
Daisy F.

### Problem 36

Find the next two terms in each sequence. Write a formula for the $n$ th term. Identify each formula as explicit or recursive.
$$4,16,64,256,1024, \dots$$

Susmith B.

### Problem 37

Find the next two terms in each sequence. Write a formula for the $n$ th term. Identify each formula as explicit or recursive.
$$49,64,81,100,121, \ldots$$

DF
Daisy F.

### Problem 38

Find the next two terms in each sequence. Write a formula for the $n$ th term. Identify each formula as explicit or recursive.
$$-1,1,-1,1,-1,1, \dots$$

Susmith B.

### Problem 39

Find the next two terms in each sequence. Write a formula for the $n$ th term. Identify each formula as explicit or recursive.
$$-16,-8,-4,-2, \ldots$$

DF
Daisy F.

### Problem 40

Find the next two terms in each sequence. Write a formula for the $n$ th term. Identify each formula as explicit or recursive.
$$-75,-68,-61,-54, \dots$$

Susmith B.

### Problem 41

Find the next two terms in each sequence. Write a formula for the $n$ th term. Identify each formula as explicit or recursive.
$$21,13,5,-3, \dots$$

DF
Daisy F.

### Problem 42

Suppose the cartoon at the right included one sheep to the left and another sheep to the right of the three shown. What "names" would you give these sheep?

Susmith B.

### Problem 43

Writing Explain the difference between a recursive formula and an explicit formula.

DF
Daisy F.

### Problem 44

a. Open-Ended Write four terms of a sequence of numbers that you can describe both recursively and explicitly.
b. Write a recursive formula and an explicit formula for your sequence.
c. Find the 20 th term of the sequence by evaluating one of your formulas. Use the other formula to check your work.

Susmith B.

### Problem 45

Use the given rule to write the $4 \mathrm{th}, 5 \mathrm{th}, 6 \mathrm{th},$ and 7 th terms of each sequence.
$$a_{1}=-1, a_{n}=a_{n-1}+n^{2}$$

DF
Daisy F.

### Problem 46

Use the given rule to write the $4 \mathrm{th}, 5 \mathrm{th}, 6 \mathrm{th},$ and 7 th terms of each sequence.
$$a_{1}=-2, a_{n}=3\left(a_{n-1}+2\right)$$

Susmith B.

### Problem 47

Use the given rule to write the $4 \mathrm{th}, 5 \mathrm{th}, 6 \mathrm{th},$ and 7 th terms of each sequence.
$$a_{n}=(n+1)^{2}$$

DF
Daisy F.

### Problem 48

Use the given rule to write the $4 \mathrm{th}, 5 \mathrm{th}, 6 \mathrm{th},$ and 7 th terms of each sequence.
$$a_{n}=2(n-1)^{3}$$

Susmith B.

### Problem 49

Use the given rule to write the $4 \mathrm{th}, 5 \mathrm{th}, 6 \mathrm{th},$ and 7 th terms of each sequence.
$$a_{n}=\frac{n^{2}}{n+1}$$

DF
Daisy F.

### Problem 50

Use the given rule to write the $4 \mathrm{th}, 5 \mathrm{th}, 6 \mathrm{th},$ and 7 th terms of each sequence.
$$a_{n}=\frac{n+1}{n+2}$$

Susmith B.

### Problem 51

Geometry Suppose you are stacking boxes in levels that form squares. The numbers of boxes in successive levels form a sequence. The figure at the left shows the top four levels as viewed from above.
a. How many boxes of equal size would you need for the next lower lever level?
b. How many boxes of equal size would you need to add three levels?
c. Suppose you are stacking a total of 285 boxes. How many levels will you have?

AG
Ankit G.

### Problem 52

Use each recursive formula to write an explicit formula for the sequence.
$$a_{1}=10, a_{n}=2 a_{n-1}$$

Susmith B.

### Problem 53

Use each recursive formula to write an explicit formula for the sequence.
$$a_{1}=-5, a_{n}=a_{n-1}-1$$

DF
Daisy F.

### Problem 54

Use each recursive formula to write an explicit formula for the sequence.
$$a_{1}=-2, a_{n}=\frac{1}{2} a_{n-1}$$

Susmith B.

### Problem 55

Use each recursive formula to write an explicit formula for the sequence.
$$a_{1}=1, a_{n}=a_{n-1}+4$$

DF
Daisy F.

### Problem 56

Finance Use the information in the ad.
a. Suppose you start a savings account at Mun e-Bank. Write both a recursive formula and an explicit formula for the amount of money you would have in the bank at the end of any week.
b. How much money would you have in the bank after four weeks?
c. Assume the bank pays interest every four weeks. To calculate your interest, multiply the balance at the end of the four weeks by 0.005. Then add that much to your account on the last day of the four-week period. Write a recursive formula for the amount of money you have after each interest payment.
d. Critical Thinking What is the bank's annual interest rate?

Susmith B.

### Problem 57

Geometry The triangular numbers form a sequence. The diagram represents the first three triangular numbers: $1,3,$ and $6 .$
a. Find the fifth and sixth triangular numbers.
b. Write a recursive formula for the $n$ th triangular number.
c. Is the explicit formula $a_{n}=\frac{1}{2}\left(n^{2}+n\right)$ the correct formula for this sequence? How do you know?

AG
Ankit G.

### Problem 58

What is the difference between the third term in the sequence whose recursive formula is $a_{1}=-5, a_{n}=2 a_{n-1}+1$ and the third term in the sequence whose recursive formula is $a_{1}=-3, a_{n}=-a_{n-1}+3 ?$
$$\begin{array}{lllll}{\text { A. } 2} & {\text { B. } 14} & {\text { C. } 20} & {\text { D. } 32}\end{array}$$

Susmith B.

### Problem 59

What is a recursive formula for the sequence whose explicit formula is $a_{n}=(n+1)^{2} ?$
F. $a_{1}=1, a_{n}=\left(a_{n-1}+1\right)^{2}$
H. $a_{1}=n, a_{n}=a_{n-1}+n$
G. $a_{1}=4, a_{n}=\left(\sqrt{a_{n-1}}+1\right)^{2}$
J. $a_{1}=n^{2}, a_{n}=\left(a_{n}-1\right)^{2}+1$

AG
Ankit G.

### Problem 60

Use the figure below for Exercises $60-62$
(GRAPH NOT COPY)
How many $1 \times 1$ squares are in the sixth term of the sequence?
$$\begin{array}{llll}{\text { A. } 21} & {\text { B. } 36} & {\text { C. } 91} & {\text { D. } 441}\end{array}$$

Susmith B.

### Problem 61

Use the figure below for Exercises $60-62$
(GRAPH NOT COPY)
Which expressions represent the first three terms of the sequence?
$$\begin{array}{ll}{\text { F. } 1^{2}, 2^{2}, 3^{2}, \ldots} & {\text { G. } 1,1+2,1+2+3, \ldots} \\ {\text { H. } 1^{2},(1+2)^{2},(1+2+3)^{2}, \ldots} & {\text { J. } 1^{2}, 1^{2}+2^{2}, 1^{2}+2^{2}+3^{2}, \ldots}\end{array}$$

AG
Ankit G.

### Problem 62

Use the figure below for Exercises $60-62$
(GRAPH NOT COPY)
Write a recursive formula for the sequence in the figure above. Explain your reasoning.

Susmith B.

### Problem 63

The graph of each equation is translated 2 units left and 3 units down. Write each new equation.
$$(x+2)^{2}+(y-1)^{2}=5$$

AG
Ankit G.

### Problem 64

The graph of each equation is translated 2 units left and 3 units down. Write each new equation.
$$\frac{(x-1)^{2}}{36}+\frac{(y-1)^{2}}{36}=1$$

Susmith B.

### Problem 65

Each point is from an inverse variation. Write an equation to model the data.
$$(1,20)$$

DF
Daisy F.

### Problem 66

Each point is from an inverse variation. Write an equation to model the data.
$$(5,2)$$

Susmith B.

### Problem 67

Each point is from an inverse variation. Write an equation to model the data.
$$(9,13)$$

DF
Daisy F.

### Problem 68

Each point is from an inverse variation. Write an equation to model the data.
$$(-3,-9)$$

Susmith B.

### Problem 69

Each point is from an inverse variation. Write an equation to model the data.
$$(2,5)$$

DF
Daisy F.

### Problem 70

Each point is from an inverse variation. Write an equation to model the data.
$$(-6,-12)$$

Susmith B.

### Problem 71

Each point is from an inverse variation. Write an equation to model the data.
$$\left(\frac{1}{2},-\frac{1}{2}\right)$$

DF
Daisy F.
$$(-10,-10)$$