Question 1

Expand the expression using the Binomial Theorem.
-$$\left( x - \frac { 3 } { \sqrt { x } } \right) ^ { 4 }$$

Accepted Answer

The answer of Expand the expression using the Binomial Theorem.
-\[\left(...

Question 2

Use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of the sequence
with the given first term, a1, and common ratio, r.
-Find $a _ { 12 }$ when $a _ { 1 } = 2,000 , r = - \frac { 1 } { 3 }$.
$A ) - \frac { 2000 } { 1594323 }$

Accepted Answer

B)  $\frac { 5989 } { 3 }$ 
C)  $\frac { 2000 } { 531441 }$ 
D)  $- \frac { 2000 } { 177147 }$ 
B)  $\frac { 5989 } { 3 }$ 
C)  $\frac { 2000 } { 531441 }$ 
D)  $- \frac { 2000 } { 177147 }$

Question 3

Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n.
-$$3 + 8 + 13 + \ldots + ( 5 n - 2 ) = \frac { n } { 2 } ( 5 n + 1 )$$

Accepted Answer

First we show that the statement is true

Question 4

Expand the expression using the Binomial Theorem.
-$$( 3 x - 2 y ) ^ { 3 }$$

Accepted Answer

A)  $9 x ^ { 3 } y - 12 x ^ { 2 } y ^ { 2 } + 4 x y ^ { 3 }$ 
B)  $27 x ^ { 3 } - 54 x ^ { 2 } y + 36 x y ^ { 2 } - 8 y ^ { 3 }$ 
C)  $9 x ^ { 3 } y - 6 x ^ { 2 } y ^ { 2 } + 4 x y ^ { 3 }$ 
D)  $27 x ^ { 3 } - 18 x ^ { 2 } y + 12 x y ^ { 2 } - 8 y ^ { 3 }$ 
A)  $9 x ^ { 3 } y - 12 x ^ { 2 } y ^ { 2 } + 4 x y ^ { 3 }$ 
B)  $27 x ^ { 3 } - 54 x ^ { 2 } y + 36 x y ^ { 2 } - 8 y ^ { 3 }$ 
C)  $9 x ^ { 3 } y - 6 x ^ { 2 } y ^ { 2 } + 4 x y ^ { 3 }$ 
D)  $27 x ^ { 3 } - 18 x ^ { 2 } y + 12 x y ^ { 2 } - 8 y ^ { 3 }$

Question 5

Find the sum.
-$$\sum _ { \mathrm { k } = 1 } ^ { 5 } \left( \frac { 1 } { 3 } \right) ( 4 ) ^ { \mathrm { k } }$$

Accepted Answer

A)  $\frac { 1,373 } { 3 }$ 
B)  $\frac { 1,385 } { 3 }$ 
C)  $\frac { 1,364 } { 3 }$ 
D)  $\frac { 1,376 } { 3 }$ 
A)  $\frac { 1,373 } { 3 }$ 
B)  $\frac { 1,385 } { 3 }$ 
C)  $\frac { 1,364 } { 3 }$ 
D)  $\frac { 1,376 } { 3 }$

Question 6

Find the sum of the sequence.
-$$\sum _ { k = 1 } ^ { 16 } ( 2 k + 7 )$$

Accepted Answer

A)  400 
B)  384 
C)  53 
D)  272 
A)  400 
B)  384 
C)  53 
D)  272

Question 7

Find the sum.
-4, 16, 64, 256, 1,024

Accepted Answer

A)  1,365 
B)  1,374 
C)  1,368 
D)  1,364 
A)  1,365 
B)  1,374 
C)  1,368 
D)  1,364

Question 8

Solve the problem.
-For the geometric sequence 64, 16, 4, 1, ... , find $a _ { n }$ .

Accepted Answer

The answer of Solve the problem.
-For the geometric sequence 64,...

Question 9

Find the nth term of the geometric sequence.
-6, 18, 54, 162, 486 A) $a _ { n } = 6 \cdot 3 n$

Accepted Answer

B)  $a _ { n } = a _ { 1 } + 3 ^ { n }$ 
C)  $a _ { n } = 6 \cdot 3 ^ { n }$ 
D)  $a _ { n } = 6 \cdot 3 ^ { n - 1 }$ 
B)  $a _ { n } = a _ { 1 } + 3 ^ { n }$ 
C)  $a _ { n } = 6 \cdot 3 ^ { n }$ 
D)  $a _ { n } = 6 \cdot 3 ^ { n - 1 }$

Question 10

Write out the sum. Do not evaluate.
-$$\sum _ { k = 1 } ^ { 4 } \left( k ^ { 2 } - 4 k - 3 \right)$$

Accepted Answer

A)  $1 - 6 - 7 - 6$ 
B)  $- 6 - 7 - 6 - 3$ 
C)  $- 6 - 7 - 6 - 3 + \ldots$ 
D)  $- 6 + 7 - 6 + 3$ 
A)  $1 - 6 - 7 - 6$ 
B)  $- 6 - 7 - 6 - 3$ 
C)  $- 6 - 7 - 6 - 3 + \ldots$ 
D)  $- 6 + 7 - 6 + 3$

Question 11

Find the indicated term using the given information.
-a = 4 , d = 4 ; a8

Accepted Answer

A)  32 
B)  36 
C)  -28 
D)  -24 
A)  32 
B)  36 
C)  -28 
D)  -24

Question 12

Write out the first five terms of the sequence.
-{4(4n - 2)}

Accepted Answer

A)  -8, 8, 24, 40, 56 
B)  2, 6, 10, 14, 18 
C)  8, 16, 24, 32, 40 
D)  8, 24, 40, 56, 72 
A)  -8, 8, 24, 40, 56 
B)  2, 6, 10, 14, 18 
C)  8, 16, 24, 32, 40 
D)  8, 24, 40, 56, 72

Question 13

Solve.
-The number of students in a school in year n is estimated by the model $$a _ { n } = 7 n ^ { 2 } + 13 n + 81$$ . About how many students are in the school in each of the first three years?

Accepted Answer

A)  108, 135, 183 
B)  101, 135, 162 
C)  101, 135, 183 
D)  114, 148, 196 
A)  108, 135, 183 
B)  101, 135, 162 
C)  101, 135, 183 
D)  114, 148, 196

Question 14

Find the sum of the arithmetic sequence.
-1 + 2 + 3 + ... + 329

Accepted Answer

A)  108,570 
B)  53956 
C)  54285 
D)  108,241 
A)  108,570 
B)  53956 
C)  54285 
D)  108,241

Question 15

Find the indicated coefficient or term.
-The 8 th term in the expansion of $( x - 2 y ) 13$

Accepted Answer

A)  $109,824 x ^ { 6 } y ^ { 8 }$ 
B)  $- 219,648 x ^ { 6 } y ^ { 7 }$ 
C)  $- 219,648 x ^ { 7 } y ^ { 6 }$ 
D)  $109,824 x ^ { 7 } y ^ { 6 }$ 
A)  $109,824 x ^ { 6 } y ^ { 8 }$ 
B)  $- 219,648 x ^ { 6 } y ^ { 7 }$ 
C)  $- 219,648 x ^ { 7 } y ^ { 6 }$ 
D)  $109,824 x ^ { 7 } y ^ { 6 }$

Question 16

Determine whether the sequence is geometric.
-3, 5, 7, 11, 13, ...

Accepted Answer

A)  Geometric 
B)  Not geometric 
A)  Geometric 
B)  Not geometric

Question 17

Evaluate the expression.
-$$\left( \begin{array} { l } 
9 \
5
\end{array} \right)$$

Accepted Answer

A)  3,024 
B)  63 
C)  126 
D)  1 
A)  3,024 
B)  63 
C)  126 
D)  1

Question 18

Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n.
-$$1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + \ldots + n ( n + 1 ) = \frac { n ( n + 1 ) ( n + 2 ) } { 3 }$$

Accepted Answer

First we show that the statement is true

Question 19

Find the indicated coefficient or term.
-The coefficient of $x ^ { 8 }$ in the expansion of $\left( x ^ { 2 } - 3 \right) ^ { 7 }$

Accepted Answer

A)  945 
B)  -945 
C)  2835 
D)  -2835 
A)  945 
B)  -945 
C)  2835 
D)  -2835

Question 20

Find the value of the annuity with the following characteristics.
-Payments of $1000 are made at the end of each year for 15 years at 9% interest compounded annually

Accepted Answer

A)  $40,472.03 
B)  $3,713.03 
C)  $29,360.92 
D)  $26,019.19 
A)  $40,472.03 
B)  $3,713.03 
C)  $29,360.92 
D)  $26,019.19

Expand the expression using the Binomial Theorem. - $\left( x - \frac { 3 } { \sqrt { x } } \right) ^ { 4 }$

Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n. - $3 + 8 + 13 + \ldots + ( 5 n - 2 ) = \frac { n } { 2 } ( 5 n + 1 )$

Expand the expression using the Binomial Theorem. - $( 3 x - 2 y ) ^ { 3 }$

Find the sum. - $\sum _ { \mathrm { k } = 1 } ^ { 5 } \left( \frac { 1 } { 3 } \right) ( 4 ) ^ { \mathrm { k } }$

Find the sum of the sequence. - $\sum _ { k = 1 } ^ { 16 } ( 2 k + 7 )$

Find the sum. -4, 16, 64, 256, 1,024

Solve the problem. -For the geometric sequence 64, 16, 4, 1, ... , find $a _ { n }$ .

Find the nth term of the geometric sequence. -6, 18, 54, 162, 486 A) $a _ { n } = 6 \cdot 3 n$ B) $a _ { n } = a _ { 1 } + 3 ^ { n }$ C) $a _ { n } = 6 \cdot 3 ^ { n }$ D) $a _ { n } = 6 \cdot 3 ^ { n - 1 }$

Write out the sum. Do not evaluate. - $\sum _ { k = 1 } ^ { 4 } \left( k ^ { 2 } - 4 k - 3 \right)$

Find the indicated term using the given information. -a = 4 , d = 4 ; a8

Write out the first five terms of the sequence. -{4(4n - 2)}

Solve. -The number of students in a school in year n is estimated by the model $a _ { n } = 7 n ^ { 2 } + 13 n + 81$ . About how many students are in the school in each of the first three years?

Find the sum of the arithmetic sequence. -1 + 2 + 3 + ... + 329

Find the indicated coefficient or term. -The 8 th term in the expansion of $( x - 2 y ) 13$

Determine whether the sequence is geometric. -3, 5, 7, 11, 13, ...

Evaluate the expression. - $\left( \begin{array} { l } 9 \\5\end{array} \right)$

Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n. - $1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + \ldots + n ( n + 1 ) = \frac { n ( n + 1 ) ( n + 2 ) } { 3 }$

Find the indicated coefficient or term. -The coefficient of $x ^ { 8 }$ in the expansion of $\left( x ^ { 2 } - 3 \right) ^ { 7 }$

Find the value of the annuity with the following characteristics. -Payments of $1000 are made at the end of each year for 15 years at 9% interest compounded annually

Functions and Their Graphs

Linear and Quadratic Functions

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Applications of Trigonometric Functions

Polar Coordinates; Vectors

Analytic Geometry

Systems of Equations and Inequalities

Counting and Probability

A Preview of Calculus: the Limit, Derivative, and Integral of a Function

Review

Filters

Exam 11: Sequences; Induction; the Binomial Theorem

Expand the expression using the Binomial Theorem. - (x−3x)4\left( x - \frac { 3 } { \sqrt { x } } \right) ^ { 4 }(x−x​3​)4

Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n. - 3+8+13+…+(5n−2)=n2(5n+1)3 + 8 + 13 + \ldots + ( 5 n - 2 ) = \frac { n } { 2 } ( 5 n + 1 )3+8+13+…+(5n−2)=2n​(5n+1)

Expand the expression using the Binomial Theorem. - (3x−2y)3( 3 x - 2 y ) ^ { 3 }(3x−2y)3

Find the sum. - ∑k=15(13)(4)k\sum _ { \mathrm { k } = 1 } ^ { 5 } \left( \frac { 1 } { 3 } \right) ( 4 ) ^ { \mathrm { k } }∑k=15​(31​)(4)k

Find the sum of the sequence. - ∑k=116(2k+7)\sum _ { k = 1 } ^ { 16 } ( 2 k + 7 )∑k=116​(2k+7)