Question 1

Find the sum of the infinite geometric series. $a + a x ^ { 2 } + a x ^ { 4 } + a x ^ { 6 } + \ldots$

Accepted Answer

The answer of Find the sum of the infinite geometric...

Question 2

Write the sum without using sigma notation. Do not evaluate. $\sum _ { n = 1 } ^ { 10 } n \cdot 2 ^ { n }$

Accepted Answer

The answer of Write the sum without using sigma notation....

Question 3

Find the $n ^ { \text {ct } }$ term of the sequence whose first several terms are $\frac { 1 } { 4 } , - \frac { 1 } { 9 } , \frac { 1 } { 16 } , - \frac { 1 } { 25 } , \ldots$

Accepted Answer

A)  $Q _ { n } = \frac { ( - 1 ) ^ { n + 1 } } { n ^ { 2 } }$ 
B)  $a _ { n } = \frac { ( - 1 ) ^ { n + 1 } } { ( n + 1 ) ^ { 2 } }$ 
C)  $a _ { \mathrm { n } } = \frac { ( n + 1 ) ^ { n } } { 4 ^ { 2 } }$ 
D)  $a _ { n } = \frac { ( 1 ) ^ { n } } { ( 2 n ) ^ { 2 } }$ 
E)  none of these 
A)  $Q _ { n } = \frac { ( - 1 ) ^ { n + 1 } } { n ^ { 2 } }$ 
B)  $a _ { n } = \frac { ( - 1 ) ^ { n + 1 } } { ( n + 1 ) ^ { 2 } }$ 
C)  $a _ { \mathrm { n } } = \frac { ( n + 1 ) ^ { n } } { 4 ^ { 2 } }$ 
D)  $a _ { n } = \frac { ( 1 ) ^ { n } } { ( 2 n ) ^ { 2 } }$ 
E)  none of these

Question 4

The first term of the arithmetic sequence is $\frac { 2} { 3 }$ and the common difference is $\left( - \frac { 2 } { 3 } \right\}$ Which term of this sequence is $- \frac { 20 } { 3 }$?

Accepted Answer

A)  $10 ^ { \mathrm { th} } \text { term }$ 
B)  $12 ^ { \text {th } } \text { term }$ 
C)  $13 ^ { \text {th } } \text { term }$ 
D)  $16 ^ { \mathrm { th } } \text { term }$ 
E)  $6 ^ { \mathrm { th} } \text { term }$ 
A)  $10 ^ { \mathrm { th} } \text { term }$ 
B)  $12 ^ { \text {th } } \text { term }$ 
C)  $13 ^ { \text {th } } \text { term }$ 
D)  $16 ^ { \mathrm { th } } \text { term }$ 
E)  $6 ^ { \mathrm { th} } \text { term }$

Question 5

Expand the expression. $$( 1 - x y ) ^ { 4 }$$

Accepted Answer

A)  $4 - x y + 6 x ^ { 2 } y ^ { 2 } - 4 x ^ { 3 } y ^ { 3 } + x ^ { 4 } y ^ { 4 }$ 
B)  $1 - x y + 6 x ^ { 2 } y ^ { 2 } - 4 x ^ { 3 } y ^ { 3 } + 4 x ^ { 4 } y ^ { 4 }$ 
C)  $1 - 4 x y + 4 x ^ { 2 } y ^ { 2 } - 6 x ^ { 3 } y ^ { 3 } + x ^ { 4 } y ^ { 4 }$ 
D)  $4 - 4 x y + x ^ { 2 } y ^ { 2 } - x ^ { 3 } y ^ { 3 } + 4 x ^ { 4 } y ^ { 4 }$ 
E)  $1 - 4 x y + 6 x ^ { 2 } y ^ { 2 } - 4 x ^ { 3 } y ^ { 3 } + x ^ { 4 } y ^ { 4 }$ 
A)  $4 - x y + 6 x ^ { 2 } y ^ { 2 } - 4 x ^ { 3 } y ^ { 3 } + x ^ { 4 } y ^ { 4 }$ 
B)  $1 - x y + 6 x ^ { 2 } y ^ { 2 } - 4 x ^ { 3 } y ^ { 3 } + 4 x ^ { 4 } y ^ { 4 }$ 
C)  $1 - 4 x y + 4 x ^ { 2 } y ^ { 2 } - 6 x ^ { 3 } y ^ { 3 } + x ^ { 4 } y ^ { 4 }$ 
D)  $4 - 4 x y + x ^ { 2 } y ^ { 2 } - x ^ { 3 } y ^ { 3 } + 4 x ^ { 4 } y ^ { 4 }$ 
E)  $1 - 4 x y + 6 x ^ { 2 } y ^ { 2 } - 4 x ^ { 3 } y ^ { 3 } + x ^ { 4 } y ^ { 4 }$

Question 6

A city was incorporated in 2004 with a population of 25,000. It is expected that the population will increase at a rate of 2% per year. The population n years after 2004 is given by the sequence $P _ { n } = 25000 ( 1.02 ) ^ { n }$ 
a) Find the first 5 terms of the sequence
b) Find the population in 2014.

Accepted Answer

The answer of A city was incorporated in 2004 with...

Question 7

The first four terms of a sequence are given. Determine whether they can be terms of an arithmetic sequence, a geometric sequence, or neither. If the sequence is arithmetic find the common difference. If the sequence is geometric find the common ratio. $1 , - \frac { 3 } { 2 } , 2 , - \frac { 5 } { 2 }$

Accepted Answer

The answer of The first four terms of a sequence...

Question 8

Write the sum without using sigma notation. $\sum _ { n = 1 } ^ { 10 } n \cdot 2 ^ { n }$

Accepted Answer

A)  $2 ^ { 1 } + 2 ^ { 2 } + 2 ^ { 3 } + \ldots + 2 ^ { 10 }$ 
B)  $1 \cdot 4 ^ { 1 } + 2 \cdot 4 ^ { 2 } + 3 \cdot 4 ^ { 3 } + \ldots + 10 \cdot 4 ^ { 10 }$ 
C)  $1 \cdot 2 ^ { 1 } + 2 \cdot 2 ^ { 2 } + 3 \cdot 2 ^ { 3 } + \ldots + 10 \cdot 2 ^ { 10 }$ 
D)  $1 \cdot 2 ^ { 1 } + 2 \cdot 3 ^ { 2 } + 3 \cdot 4 ^ { 3 } + \ldots + 10 \cdot 2 ^ { 10 }$ 
E)  none of these 
A)  $2 ^ { 1 } + 2 ^ { 2 } + 2 ^ { 3 } + \ldots + 2 ^ { 10 }$ 
B)  $1 \cdot 4 ^ { 1 } + 2 \cdot 4 ^ { 2 } + 3 \cdot 4 ^ { 3 } + \ldots + 10 \cdot 4 ^ { 10 }$ 
C)  $1 \cdot 2 ^ { 1 } + 2 \cdot 2 ^ { 2 } + 3 \cdot 2 ^ { 3 } + \ldots + 10 \cdot 2 ^ { 10 }$ 
D)  $1 \cdot 2 ^ { 1 } + 2 \cdot 3 ^ { 2 } + 3 \cdot 4 ^ { 3 } + \ldots + 10 \cdot 2 ^ { 10 }$ 
E)  none of these

Question 9

The first term of a geometric sequence is $12$ and the second term is $4$ .
 Find the fifth term.

Accepted Answer

The answer of The first term of a geometric sequence...

Question 10

Find the first four terms sequence $a _ { n } = n - 1$

Accepted Answer

A)  $a _ { 1 } = - 1$, $a _ { 2 } = 0$, $a _ { 3 } = 1$, $a _ { 4 } = 2$ 
B)  $a _ { 1 } = 0$, $a _ { 2 } = 1$, $a = 2$, $a _ { 4 } = 3$ 
C)  $a _ { 1 } = - 2$, $a _ { 2 } = - 3$, $a _ { 3 } = - 4$, $a _ { 4 } = - 5$ 
D)  $a _ { 1 } = 1$, $a _ { 2 } = 2$, $a = 3$, $a _ { 4 } = 4$ 
E) none of these 
A)  $a _ { 1 } = - 1$, $a _ { 2 } = 0$, $a _ { 3 } = 1$, $a _ { 4 } = 2$ 
B)  $a _ { 1 } = 0$, $a _ { 2 } = 1$, $a = 2$, $a _ { 4 } = 3$ 
C)  $a _ { 1 } = - 2$, $a _ { 2 } = - 3$, $a _ { 3 } = - 4$, $a _ { 4 } = - 5$ 
D)  $a _ { 1 } = 1$, $a _ { 2 } = 2$, $a = 3$, $a _ { 4 } = 4$ 
E) none of these

Question 11

Find the fourth term and the $n ^ { t h }$ term of the geometric sequence given $\alpha = 7$ 
and $r= \frac { 1 } { 7 }$

Accepted Answer

The answer of Find the fourth term and the \(n...

Question 12

Determine whether the expression is a partial sum of an arithmetic or geometric sequence. Then find the sum. $2 + 4 + 6 + 8 + \dots+ 100$

Accepted Answer

The answer of Determine whether the expression is a partial...

Question 13

Find the first four terms and the $10 ^ { \mathrm { ct } }$ 
term of the sequence $a _ { n } = n - 1$ 
.

Accepted Answer

The answer of Find the first four terms and the...

Question 14

Find the fourth term of the geometric sequence given $a _ { 1 } = 7$ and $\boldsymbol { r } = \frac { 1 } { 7 }$

Accepted Answer

A)  $a _ { 4 } = \frac { 1 } { 7 }$ 
B)  $a _ { 4 } = - \frac { 1 } { 4 g }$ 
C)  $a _ { 4 } = \frac { 1 } { 14 }$ 
D)  $a _ { 4 } = \frac { 1 } { 49 }$ 
E)  $a _ { 4 } = 49$ 
A)  $a _ { 4 } = \frac { 1 } { 7 }$ 
B)  $a _ { 4 } = - \frac { 1 } { 4 g }$ 
C)  $a _ { 4 } = \frac { 1 } { 14 }$ 
D)  $a _ { 4 } = \frac { 1 } { 49 }$ 
E)  $a _ { 4 } = 49$

Question 15

Find the $1000 ^ { \text {th } }$ term of the sequence $a _ { n } = ( - 1 ) ^ { n } \frac { n + 2 } { n }$

Accepted Answer

A)  $a _ { 1000 } = - \frac { 501 } { 500 }$ 
B)  $a _ { 1000 } = \frac { 251 } { 250 }$ 
C)  $a _ { 1000 } = \frac { 126 } { 125 }$ 
D)  $a _ { 1000 } = - \frac { 126 } { 125 }$ 
E)  $a _ { 1000 } = \frac { 501 } { 500 }$ 
A)  $a _ { 1000 } = - \frac { 501 } { 500 }$ 
B)  $a _ { 1000 } = \frac { 251 } { 250 }$ 
C)  $a _ { 1000 } = \frac { 126 } { 125 }$ 
D)  $a _ { 1000 } = - \frac { 126 } { 125 }$ 
E)  $a _ { 1000 } = \frac { 501 } { 500 }$

Question 16

Find the sum of the infinite geometric series. $\sum _ { n = 1 } ^ { \infty } 3 \left( \frac { 1 } { 2 } \right) ^ { n - 1 }$

Accepted Answer

A)  $6$ 
B)  $3$ 
C)  $12$ 
D)  $9$ 
E)  $18$ 
A)  $6$ 
B)  $3$ 
C)  $12$ 
D)  $9$ 
E)  $18$

Question 17

Find the sum. $2 + 4 + 6 + 8 + \dots + 100$

Accepted Answer

A)  $2000$ 
B)  $200$ 
C)  $2050$ 
D)  $1550$ 
E)  none of these 
A)  $2000$ 
B)  $200$ 
C)  $2050$ 
D)  $1550$ 
E)  none of these

Question 18

Find the sum of the infinite geometric series. $x + x y ^ { 2 } + x y ^ { 4 } + x y ^ { 6 } + \ldots$

Accepted Answer

The answer of Find the sum of the infinite geometric...

Question 19

Write the sum using sigma notation. $\frac { 1 } { 1 \cdot 2 } + \frac { 1 } { 2 \cdot 3 } + \frac { 1 } { 3 \cdot 4 } + \ldots + \frac { 1 } { 999 \cdot 1000 }$

Accepted Answer

A)  $\sum _ { n = 1 } ^ { 1000 } \frac { 1 } { n ( n + 1 ) }$ 
B)  $\sum _ { n = 1 } ^ { 1000 } \frac { 1 } { n ( n - 1 ) }$ 
C)  $\sum _ { n = 1 } ^ { 999 } \frac { 1 } { n ( n + 1 ) }$ 
D)  $\sum _ { n = 1 } ^ { 1001 } \frac { 1 } { n ( n + 1 ) }$ 
E)  $\sum _ { n = 1 } ^ { 999 } \frac { 1 } { n ( n - 1 ) }$ 
A)  $\sum _ { n = 1 } ^ { 1000 } \frac { 1 } { n ( n + 1 ) }$ 
B)  $\sum _ { n = 1 } ^ { 1000 } \frac { 1 } { n ( n - 1 ) }$ 
C)  $\sum _ { n = 1 } ^ { 999 } \frac { 1 } { n ( n + 1 ) }$ 
D)  $\sum _ { n = 1 } ^ { 1001 } \frac { 1 } { n ( n + 1 ) }$ 
E)  $\sum _ { n = 1 } ^ { 999 } \frac { 1 } { n ( n - 1 ) }$

Question 20

Find the first five terms of the sequence $a _ { n } = 3 a _ { n - 1 } - 1 , \text { where } a _ { 1 } = 3$

Accepted Answer

A)  $a _ { 1 } = 1$, $a _ { 2 } = 6$, $a _ { 1 } = 21$, $a _ { 4 } = 66$, $a _ { 3 } = 201$ 
B)  $a _ { 1 } = 3$, $a _ { 2 } = 9$, $a _ { \jmath } = 27$, $a _ { 4 } = 69$, $a _ { 3 } = 226$ 
C)  $a _ { 1 } = 3$, $a _ { 2 } = 8$, $a _ { 3 } = 23$, $a _ { 4 } = 68$, $a _ { 3 } = 203$ 
D)  $a _ { 1 } = 1$, $a _ { 2 } = 7$, $a _ { 3 } = 22$, $a _ { 4 } = 67$, $a _ { 3 } = 202$ 
E)  none of these 
A)  $a _ { 1 } = 1$, $a _ { 2 } = 6$, $a _ { 1 } = 21$, $a _ { 4 } = 66$, $a _ { 3 } = 201$ 
B)  $a _ { 1 } = 3$, $a _ { 2 } = 9$, $a _ { \jmath } = 27$, $a _ { 4 } = 69$, $a _ { 3 } = 226$ 
C)  $a _ { 1 } = 3$, $a _ { 2 } = 8$, $a _ { 3 } = 23$, $a _ { 4 } = 68$, $a _ { 3 } = 203$ 
D)  $a _ { 1 } = 1$, $a _ { 2 } = 7$, $a _ { 3 } = 22$, $a _ { 4 } = 67$, $a _ { 3 } = 202$ 
E)  none of these

Find the sum of the infinite geometric series. $a + a x ^ { 2 } + a x ^ { 4 } + a x ^ { 6 } + \ldots$

Write the sum without using sigma notation. Do not evaluate. $\sum _ { n = 1 } ^ { 10 } n \cdot 2 ^ { n }$

Find the $n ^ { \text {ct } }$ term of the sequence whose first several terms are $\frac { 1 } { 4 } , - \frac { 1 } { 9 } , \frac { 1 } { 16 } , - \frac { 1 } { 25 } , \ldots$

The first term of the arithmetic sequence is $\frac { 2} { 3 }$ and the common difference is $\left( - \frac { 2 } { 3 } \right\}$ Which term of this sequence is $- \frac { 20 } { 3 }$ ?

Expand the expression. $( 1 - x y ) ^ { 4 }$

Write the sum without using sigma notation. $\sum _ { n = 1 } ^ { 10 } n \cdot 2 ^ { n }$

The first term of a geometric sequence is $12$ and the second term is $4$ . Find the fifth term.

Find the first four terms sequence $a _ { n } = n - 1$

Find the fourth term and the $n ^ { t h }$ term of the geometric sequence given $\alpha = 7$ and $r= \frac { 1 } { 7 }$

Determine whether the expression is a partial sum of an arithmetic or geometric sequence. Then find the sum. $2 + 4 + 6 + 8 + \dots+ 100$

Find the first four terms and the $10 ^ { \mathrm { ct } }$ term of the sequence $a _ { n } = n - 1$ .

Find the fourth term of the geometric sequence given $a _ { 1 } = 7$ and $\boldsymbol { r } = \frac { 1 } { 7 }$

Find the $1000 ^ { \text {th } }$ term of the sequence $a _ { n } = ( - 1 ) ^ { n } \frac { n + 2 } { n }$

Find the sum of the infinite geometric series. $\sum _ { n = 1 } ^ { \infty } 3 \left( \frac { 1 } { 2 } \right) ^ { n - 1 }$

Find the sum. $2 + 4 + 6 + 8 + \dots + 100$

Find the sum of the infinite geometric series. $x + x y ^ { 2 } + x y ^ { 4 } + x y ^ { 6 } + \ldots$

Write the sum using sigma notation. $\frac { 1 } { 1 \cdot 2 } + \frac { 1 } { 2 \cdot 3 } + \frac { 1 } { 3 \cdot 4 } + \ldots + \frac { 1 } { 999 \cdot 1000 }$

Find the first five terms of the sequence $a _ { n } = 3 a _ { n - 1 } - 1 , \text { where } a _ { 1 } = 3$

Equations and Graphs

Functions

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions: Right Triangle Approach

Trigonometric Functions: Unit Circle Approach

Conic Sections

Polar Coordinates and Parametric Equations

Vectors in Two and Three Dimensions

Systems of Equations and Inequalities

Matrices and Determinants

Conic Sections

Counting and Probability

Filters

Exam 13: Sequences and Series

Find the sum of the infinite geometric series. a+ax2+ax4+ax6+…a + a x ^ { 2 } + a x ^ { 4 } + a x ^ { 6 } + \ldotsa+ax2+ax4+ax6+…

Write the sum without using sigma notation. Do not evaluate. ∑n=110n⋅2n\sum _ { n = 1 } ^ { 10 } n \cdot 2 ^ { n }∑n=110​n⋅2n

Find the nct n ^ { \text {ct } }nct term of the sequence whose first several terms are 14,−19,116,−125,…\frac { 1 } { 4 } , - \frac { 1 } { 9 } , \frac { 1 } { 16 } , - \frac { 1 } { 25 } , \ldots41​,−91​,161​,−251​,…

The first term of the arithmetic sequence is 23\frac { 2} { 3 }32​ and the common difference is (−23}\left( - \frac { 2 } { 3 } \right\}(−32​} Which term of this sequence is −203- \frac { 20 } { 3 }−320​ ?

Expand the expression. (1−xy)4( 1 - x y ) ^ { 4 }(1−xy)4

Write the sum without using sigma notation. ∑n=110n⋅2n\sum _ { n = 1 } ^ { 10 } n \cdot 2 ^ { n }∑n=110​n⋅2n

The first term of a geometric sequence is 121212 and the second term is 444 . Find the fifth term.

Find the first four terms sequence an=n−1a _ { n } = n - 1an​=n−1

Find the fourth term and the nthn ^ { t h }nth term of the geometric sequence given α=7\alpha = 7α=7 and r=17r= \frac { 1 } { 7 }r=71​

Determine whether the expression is a partial sum of an arithmetic or geometric sequence. Then find the sum. 2+4+6+8+⋯+1002 + 4 + 6 + 8 + \dots+ 1002+4+6+8+⋯+100

Find the first four terms and the 10ct10 ^ { \mathrm { ct } }10ct term of the sequence an=n−1a _ { n } = n - 1an​=n−1 .

Find the fourth term of the geometric sequence given a1=7a _ { 1 } = 7a1​=7 and r=17\boldsymbol { r } = \frac { 1 } { 7 }r=71​

Find the 1000th 1000 ^ { \text {th } }1000th term of the sequence an=(−1)nn+2na _ { n } = ( - 1 ) ^ { n } \frac { n + 2 } { n }an​=(−1)nnn+2​

Find the sum of the infinite geometric series. ∑n=1∞3(12)n−1\sum _ { n = 1 } ^ { \infty } 3 \left( \frac { 1 } { 2 } \right) ^ { n - 1 }∑n=1∞​3(21​)n−1

Find the sum. 2+4+6+8+⋯+1002 + 4 + 6 + 8 + \dots + 1002+4+6+8+⋯+100

Find the sum of the infinite geometric series. x+xy2+xy4+xy6+…x + x y ^ { 2 } + x y ^ { 4 } + x y ^ { 6 } + \ldotsx+xy2+xy4+xy6+…

Write the sum using sigma notation. 11⋅2+12⋅3+13⋅4+…+1999⋅1000\frac { 1 } { 1 \cdot 2 } + \frac { 1 } { 2 \cdot 3 } + \frac { 1 } { 3 \cdot 4 } + \ldots + \frac { 1 } { 999 \cdot 1000 }1⋅21​+2⋅31​+3⋅41​+…+999⋅10001​

Find the first five terms of the sequence an=3an−1−1, where a1=3a _ { n } = 3 a _ { n - 1 } - 1 , \text { where } a _ { 1 } = 3an​=3an−1​−1, where a1​=3

Equations and Graphs

Functions

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions: Right Triangle Approach

Trigonometric Functions: Unit Circle Approach

Conic Sections

Polar Coordinates and Parametric Equations

Vectors in Two and Three Dimensions

Systems of Equations and Inequalities

Matrices and Determinants

Conic Sections

Counting and Probability

Filters

Find the sum of the infinite geometric series. $a + a x ^ { 2 } + a x ^ { 4 } + a x ^ { 6 } + \ldots$

Write the sum without using sigma notation. Do not evaluate. $\sum _ { n = 1 } ^ { 10 } n \cdot 2 ^ { n }$

Find the $n ^ { \text {ct } }$ term of the sequence whose first several terms are $\frac { 1 } { 4 } , - \frac { 1 } { 9 } , \frac { 1 } { 16 } , - \frac { 1 } { 25 } , \ldots$

The first term of the arithmetic sequence is $\frac { 2} { 3 }$ and the common difference is $\left( - \frac { 2 } { 3 } \right\}$ Which term of this sequence is $- \frac { 20 } { 3 }$ ?

Expand the expression. $( 1 - x y ) ^ { 4 }$

Write the sum without using sigma notation. $\sum _ { n = 1 } ^ { 10 } n \cdot 2 ^ { n }$

The first term of a geometric sequence is $12$ and the second term is $4$ . Find the fifth term.

Find the first four terms sequence $a _ { n } = n - 1$

Find the fourth term and the $n ^ { t h }$ term of the geometric sequence given $\alpha = 7$ and $r= \frac { 1 } { 7 }$

Determine whether the expression is a partial sum of an arithmetic or geometric sequence. Then find the sum. $2 + 4 + 6 + 8 + \dots+ 100$

Find the first four terms and the $10 ^ { \mathrm { ct } }$ term of the sequence $a _ { n } = n - 1$ .

Find the fourth term of the geometric sequence given $a _ { 1 } = 7$ and $\boldsymbol { r } = \frac { 1 } { 7 }$

Find the $1000 ^ { \text {th } }$ term of the sequence $a _ { n } = ( - 1 ) ^ { n } \frac { n + 2 } { n }$

Find the sum of the infinite geometric series. $\sum _ { n = 1 } ^ { \infty } 3 \left( \frac { 1 } { 2 } \right) ^ { n - 1 }$

Find the sum. $2 + 4 + 6 + 8 + \dots + 100$

Find the sum of the infinite geometric series. $x + x y ^ { 2 } + x y ^ { 4 } + x y ^ { 6 } + \ldots$

Write the sum using sigma notation. $\frac { 1 } { 1 \cdot 2 } + \frac { 1 } { 2 \cdot 3 } + \frac { 1 } { 3 \cdot 4 } + \ldots + \frac { 1 } { 999 \cdot 1000 }$

Find the first five terms of the sequence $a _ { n } = 3 a _ { n - 1 } - 1 , \text { where } a _ { 1 } = 3$