Describe each pattern formed. Find the next three terms.

$$

80,77,74,71,68, \dots

$$

Daisy F.

Numerade Educator

Describe each pattern formed. Find the next three terms.

$$

4,8,16,32,64, \dots

$$

Susmith B.

Numerade Educator

Describe each pattern formed. Find the next three terms.

$$

0,3,7,12,18, \dots

$$

Ankit G.

Numerade Educator

Describe each pattern formed. Find the next three terms.

$$

1,4,7,10,13, \dots

$$

Susmith B.

Numerade Educator

Describe each pattern formed. Find the next three terms.

$$

100,10,1,0.1,0.01, \dots

$$

Sarah W.

Numerade Educator

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.

Numerade Educator

Describe each pattern formed. Find the next three terms.

$$

4,-8,16,-32,64, \ldots

$$

Daisy F.

Numerade Educator

Describe each pattern formed. Find the next three terms.

$$

1,2,6,24,120, \dots

$$

Susmith B.

Numerade Educator

Describe each pattern formed. Find the next three terms.

$$

0,1,0, \frac{1}{3}, 0, \frac{1}{5}, \dots

$$

Daisy F.

Numerade Educator

Fractal Geometry Draw the first four figures of the sequence described.

_____ is replaced by

Ankit G.

Numerade Educator

Fractal Geometry Draw the first four figures of the sequence described.

(TRIANGLE NOT COPY) is replaced by (TRIANGLE NOT COPY)

Ankit G.

Numerade Educator

Write a recursive formula for each sequence. Then find the next term.

$$

-2,-1,0,1,2, \ldots

$$

Susmith B.

Numerade Educator

Write a recursive formula for each sequence. Then find the next term.

$$

43,41,39,37,35, \ldots

$$

Daisy F.

Numerade Educator

Write a recursive formula for each sequence. Then find the next term.

$$

40,20,10,5, \frac{5}{2}, \dots

$$

Susmith B.

Numerade Educator

Write a recursive formula for each sequence. Then find the next term.

$$

6,1,-4,-9, \dots

$$

Daisy F.

Numerade Educator

Write a recursive formula for each sequence. Then find the next term.

$$

144,36,9, \frac{9}{4}, \dots

$$

Susmith B.

Numerade Educator

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

$$

Daisy F.

Numerade Educator

Write an explicit formula for each sequence. Then find $a_{12}$

$$

4,5,6,7,8, \dots

$$

Susmith B.

Numerade Educator

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

$$

Daisy F.

Numerade Educator

Write an explicit formula for each sequence. Then find $a_{12}$

$$

4,7,10,13,16, \dots

$$

Susmith B.

Numerade Educator

Write an explicit formula for each sequence. Then find $a_{12}$

$$

3,7,11,15,19, \ldots

$$

Daisy F.

Numerade Educator

Write an explicit formula for each sequence. Then find $a_{12}$

$$

-2 \frac{1}{2},-2,-1 \frac{1}{2},-1, \ldots

$$

Susmith B.

Numerade Educator

Write an explicit formula for each sequence. Then find $a_{12}$

$$

2,5,10,17,26, \dots

$$

Daisy F.

Numerade Educator

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.

Numerade Educator

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)

$$

Daisy F.

Numerade Educator

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.

Numerade Educator

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

$$

Daisy F.

Numerade Educator

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.

Numerade Educator

Decide whether each formula is explicit or recursive. Then find the first five terms of each sequence.

$$

a_{n}=2 n^{2}+1

$$

Daisy F.

Numerade Educator

Decide whether each formula is explicit or recursive. Then find the first five terms of each sequence.

$$

a_{n}=5 n

$$

Susmith B.

Numerade Educator

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

$$

Daisy F.

Numerade Educator

Entertainment Suppose you are building a tower of cards with levels as displayed below. Complete the

table, assuming the pattern continues.

Susmith B.

Numerade Educator

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

$$

Daisy F.

Numerade Educator

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.

Numerade Educator

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

$$

Daisy F.

Numerade Educator

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.

Numerade Educator

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

$$

Daisy F.

Numerade Educator

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.

Numerade Educator

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

$$

Daisy F.

Numerade Educator

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.

Numerade Educator

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

$$

Daisy F.

Numerade Educator

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.

Numerade Educator

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

Daisy F.

Numerade Educator

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.

Numerade Educator

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}

$$

Daisy F.

Numerade Educator

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.

Numerade Educator

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}

$$

Daisy F.

Numerade Educator

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.

Numerade Educator

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}

$$

Daisy F.

Numerade Educator

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.

Numerade Educator

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?

Ankit G.

Numerade Educator

Use each recursive formula to write an explicit formula for the sequence.

$$

a_{1}=10, a_{n}=2 a_{n-1}

$$

Susmith B.

Numerade Educator

Use each recursive formula to write an explicit formula for the sequence.

$$

a_{1}=-5, a_{n}=a_{n-1}-1

$$

Daisy F.

Numerade Educator

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.

Numerade Educator

Use each recursive formula to write an explicit formula for the sequence.

$$

a_{1}=1, a_{n}=a_{n-1}+4

$$

Daisy F.

Numerade Educator

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.

Numerade Educator

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?

Ankit G.

Numerade Educator

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.

Numerade Educator

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. $a_{1}=n^{2}, a_{n}=\left(a_{n}-1\right)^{2}+1$

Ankit G.

Numerade Educator

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.

Numerade Educator

Use the figure below for Exercises $60-62$

(GRAPH NOT COPY)

Which expressions represent the first three terms of the sequence?

$$\begin{array}{ll}{\text { F. } 1^{2}, 2^{2}, 3^{2}, \ldots} & {\text { G. } 1,1+2,1+2+3, \ldots} \\ {\text { H. } 1^{2},(1+2)^{2},(1+2+3)^{2}, \ldots} & {\text { J. } 1^{2}, 1^{2}+2^{2}, 1^{2}+2^{2}+3^{2}, \ldots}\end{array}$$

Ankit G.

Numerade Educator

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.

Numerade Educator

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

$$

Ankit G.

Numerade Educator

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.

Numerade Educator

Each point is from an inverse variation. Write an equation to model the data.

$$

(1,20)

$$

Daisy F.

Numerade Educator

Each point is from an inverse variation. Write an equation to model the data.

$$

(5,2)

$$

Susmith B.

Numerade Educator

Each point is from an inverse variation. Write an equation to model the data.

$$

(9,13)

$$

Daisy F.

Numerade Educator

Each point is from an inverse variation. Write an equation to model the data.

$$

(-3,-9)

$$

Susmith B.

Numerade Educator

Each point is from an inverse variation. Write an equation to model the data.

$$

(2,5)

$$

Daisy F.

Numerade Educator

Each point is from an inverse variation. Write an equation to model the data.

$$

(-6,-12)

$$

Susmith B.

Numerade Educator

Each point is from an inverse variation. Write an equation to model the data.

$$

\left(\frac{1}{2},-\frac{1}{2}\right)

$$

Daisy F.

Numerade Educator

Each point is from an inverse variation. Write an equation to model the data.

$$

(-10,-10)

$$

Susmith B.

Numerade Educator