Question 1

Find the sum of the sequence.
-$$\sum _ { k = 4 } ^ { 7 } 6 k$$

Accepted Answer

A)  132 
B)  90 
C)  42 
D)  66 
A)  132 
B)  90 
C)  42 
D)  66

Question 2

Write out the sum. Do not evaluate.
-$$\sum _ { k = 1 } ^ { n } 2 ^ { k + 1 }$$

Accepted Answer

A)  $2 ^ { 2 } + 2 ^ { 3 } + 2 ^ { 4 } + \ldots + 2 ^ { n + 1 }$ 
B)  $2 + 2 ^ { 2 } + 2 ^ { 3 } + \ldots + 2 ^ { n + 1 }$ 
C)  $2 + 2 ^ { 2 } + 2 ^ { 3 } + \ldots + 2 ^ { n }$ 
D)  $2 ^ { 2 } + 2 ^ { 3 } + 2 ^ { 4 } + \ldots + 2 ^ { n }$ 
A)  $2 ^ { 2 } + 2 ^ { 3 } + 2 ^ { 4 } + \ldots + 2 ^ { n + 1 }$ 
B)  $2 + 2 ^ { 2 } + 2 ^ { 3 } + \ldots + 2 ^ { n + 1 }$ 
C)  $2 + 2 ^ { 2 } + 2 ^ { 3 } + \ldots + 2 ^ { n }$ 
D)  $2 ^ { 2 } + 2 ^ { 3 } + 2 ^ { 4 } + \ldots + 2 ^ { n }$

Question 3

Write out the first five terms of the sequence.
-$$\left\{ 2 ^ { n } \right\}$$

Accepted Answer

A)  2, 4, 8, 16, 32 
B)  1, 2, 4, 8, 16 
C)  4, 8, 16, 32, 64 
D)  1, 4, 9, 16, 25 
A)  2, 4, 8, 16, 32 
B)  1, 2, 4, 8, 16 
C)  4, 8, 16, 32, 64 
D)  1, 4, 9, 16, 25

Question 4

Find the indicated coefficient or term.
-The 12th term in the expansion of $( x - 2 y ) ^ { 13 }$

Accepted Answer

A)  $79,872 x 11 y ^ { 2 }$ 
B)  $79,872 x ^ { 2 } y 12$ 
C)  $- 159,744 x ^ { 11 } y ^ { 2 }$ 
D)  $- 159,744 x ^ { 2 } y 11$ 
A)  $79,872 x 11 y ^ { 2 }$ 
B)  $79,872 x ^ { 2 } y 12$ 
C)  $- 159,744 x ^ { 11 } y ^ { 2 }$ 
D)  $- 159,744 x ^ { 2 } y 11$

Question 5

Express the sum using summation notation.
-$$4 ^ { 2 } + 8 ^ { 3 } + 12 ^ { 4 } + \ldots + 32 ^ { 9 }$$

Accepted Answer

A)  $\sum _ { \mathrm { k } = 1 } ^ { 8 } 4 \mathrm { k } ^ { 2 \mathrm { k } - 1 }$ 
B)  $\sum _ { k = 1 } ^ { 8 } ( 4 k ) ^ { k }$ 
C)  $\sum _ { k = 1 } ^ { 8 } 2 ( k - 1 ) ^ { \mathrm { k } + 1 }$ 
D)  $\sum _ { k = 1 } ^ { 8 } ( 4 k ) ^ { k + 1 }$ 
A)  $\sum _ { \mathrm { k } = 1 } ^ { 8 } 4 \mathrm { k } ^ { 2 \mathrm { k } - 1 }$ 
B)  $\sum _ { k = 1 } ^ { 8 } ( 4 k ) ^ { k }$ 
C)  $\sum _ { k = 1 } ^ { 8 } 2 ( k - 1 ) ^ { \mathrm { k } + 1 }$ 
D)  $\sum _ { k = 1 } ^ { 8 } ( 4 k ) ^ { k + 1 }$

Question 6

Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n.
-$$\text { Show that the statement "n } { } ^ { 2 } - n + 3 \text { is a prime number" is true for } n = 1 \text {, but is not true for } n = 3 \text {. }$$

Accepted Answer

When @#LAT-DLM&, which is a prime number

Question 7

Solve the problem.
-Jennifer takes a job with a starting salary of $40,000 for the first year with an annual increase of 4% beginning in the second year. What is Jennifer's salary, to the nearest dollar, in the sixth year?

Accepted Answer

A)  $48,666 
B)  $50,891 
C)  $49,664 
D)  $47,367 
A)  $48,666 
B)  $50,891 
C)  $49,664 
D)  $47,367

Question 8

Solve the problem.
-A bicycle wheel rotates 500 times in a minute as long as the rider is pedalling. If the rider stops pedalling, the wheel starts to slow down. Each minute it will rotate only $4 / 5$ as many times as in the preceding minute. How
Many times will the wheel rotate in the 4th minute after the rider's feet leave the pedals? Round your answer to
The nearest unit.

Accepted Answer

A)  0 times 
B)  158 times 
C)  211 times 
D)  8 times 
A)  0 times 
B)  158 times 
C)  211 times 
D)  8 times

Question 9

Expand the expression using the Binomial Theorem.
-$( x + 1 ) ^ { 6 }$

Accepted Answer

A)  $x ^ { 6 } + 6 x ^ { 5 } + 15 x ^ { 4 } + 20 x ^ { 3 } + 15 x ^ { 2 } + 6 x + 6$ 
B)  $x ^ { 6 } + 6 x ^ { 5 } + 15 x ^ { 4 } + 20 x ^ { 3 } + 15 x ^ { 2 } + 6 x + 1$ 
C)  $x ^ { 6 } + 6 x ^ { 5 } + 30 x ^ { 4 } + 120 x ^ { 3 } + 30 x ^ { 2 } + 6 x + 1$ 
D)  $x ^ { 6 } + 6 x ^ { 5 } + 30 x ^ { 4 } + 120 x ^ { 3 } + 360 x ^ { 2 } + 720 x + 720$ 
A)  $x ^ { 6 } + 6 x ^ { 5 } + 15 x ^ { 4 } + 20 x ^ { 3 } + 15 x ^ { 2 } + 6 x + 6$ 
B)  $x ^ { 6 } + 6 x ^ { 5 } + 15 x ^ { 4 } + 20 x ^ { 3 } + 15 x ^ { 2 } + 6 x + 1$ 
C)  $x ^ { 6 } + 6 x ^ { 5 } + 30 x ^ { 4 } + 120 x ^ { 3 } + 30 x ^ { 2 } + 6 x + 1$ 
D)  $x ^ { 6 } + 6 x ^ { 5 } + 30 x ^ { 4 } + 120 x ^ { 3 } + 360 x ^ { 2 } + 720 x + 720$

Question 10

Express the sum using summation notation with a lower limit of summation not necessarily 1 and with k for the index of
summation.
-$$4 + \frac { 9 } { 2 } + 5 + \frac { 11 } { 2 } + \ldots + \frac { 15 } { 2 }$$

Accepted Answer

A)  $\sum _ { k = 8 } ^ { 15 } \frac { k } { 2 }$ 
B)  $\sum _ { k = 8 } ^ { 12 } \frac { \mathrm { k } } { 2 }$ 
C)  $\sum _ { \mathrm { k } = 1 } ^ { 15 } \frac { \mathrm { k } } { 2 }$ 
D)  $\sum _ { k = 2 } ^ { 15 } \frac { \mathrm { k } } { 2 }$ 
A)  $\sum _ { k = 8 } ^ { 15 } \frac { k } { 2 }$ 
B)  $\sum _ { k = 8 } ^ { 12 } \frac { \mathrm { k } } { 2 }$ 
C)  $\sum _ { \mathrm { k } = 1 } ^ { 15 } \frac { \mathrm { k } } { 2 }$ 
D)  $\sum _ { k = 2 } ^ { 15 } \frac { \mathrm { k } } { 2 }$

Question 11

Determine whether the given sequence is arithmetic, geometric, or neither. If arithmetic, find the common difference. If
geometric, find the common ratio.
-$\left\{ 3 n ^ { 2 } \right\}$

Accepted Answer

A)  Geometric, $r = 3$ 
B)  Arithmetic, $\mathrm { d } = 3$ 
C)  Neither 
D)  Geometric, $r = \frac { 3 } { 2 }$ 
A)  Geometric, $r = 3$ 
B)  Arithmetic, $\mathrm { d } = 3$ 
C)  Neither 
D)  Geometric, $r = \frac { 3 } { 2 }$

Question 12

Find the value of the annuity with the following characteristics.
-Payments of $100 are made at the end of each year for 7 years at 4% interest compounded annually

Accepted Answer

A)  $3,289.83 
B)  $789.83 
C)  $663.30 
D)  $316.33 
A)  $3,289.83 
B)  $789.83 
C)  $663.30 
D)  $316.33

Question 13

Find the indicated coefficient or term.
-The 3rd term in the expansion of ( $( 7 x + 9 ) ^ { 3 }$

Accepted Answer

A)  3,402x 
B)  1,323x2 
C)  1,701x 
D)  81 
A)  3,402x 
B)  1,323x2 
C)  1,701x 
D)  81

Question 14

Determine whether the infinite geometric series converges or diverges. If it converges, find its sum.
-$$1 + \frac { 1 } { 2 } + \frac { 1 } { 4 } + \cdots$$

Accepted Answer

A)  Converges; 2 
B)  Converges; 1 
C)  Converges; $\frac { 3 } { 2 }$ 
D)  Converges; $\frac { 1 } { 2 }$ 
A)  Converges; 2 
B)  Converges; 1 
C)  Converges; $\frac { 3 } { 2 }$ 
D)  Converges; $\frac { 1 } { 2 }$

Question 15

Determine whether the infinite geometric series converges or diverges. If it converges, find its sum.
-$\sum _ { k = 1 } ^ { \infty } 4 \left( \frac { 2 } { 3 } \right) ^ { k - 1 }$

Accepted Answer

A)  Converges; 16 
B)  Converges; 4 
C)  Converges; 12 
D)  Converges; $\frac { 8 } { 3 }$ 
A)  Converges; 16 
B)  Converges; 4 
C)  Converges; 12 
D)  Converges; $\frac { 8 } { 3 }$

Question 16

Express the sum using summation notation.
-$$\frac { 5 ^ { 14 } } { 9 ^ { 4 } } + \frac { 5 ^ { 13 } } { 9 ^ { 5 } } + \frac { 5 ^ { 12 } } { 9 ^ { 6 } } + \ldots + \frac { 5 ^ { 7 } } { 9 ^ { 11 } }$$

Accepted Answer

A)  $\sum _ { k = 1 } ^ { 4 } \frac { 5 ^ { 15 + k } } { 9 ^ { 3 + k } }$ 
B)  $\sum _ { k = 1 } ^ { 8 } \frac { 5 ^ { - k } } { 9 ^ { 3 + k } }$ 
C)  $\sum _ { k = 1 } ^ { 8 } \frac { 5 ^ { 15 - k } } { 9 ^ { k } }$ 
D)  $\sum _ { k = 1 } ^ { 8 } \frac { 5 ^ { 15 - k } } { 9 ^ { 3 + k } }$ 
A)  $\sum _ { k = 1 } ^ { 4 } \frac { 5 ^ { 15 + k } } { 9 ^ { 3 + k } }$ 
B)  $\sum _ { k = 1 } ^ { 8 } \frac { 5 ^ { - k } } { 9 ^ { 3 + k } }$ 
C)  $\sum _ { k = 1 } ^ { 8 } \frac { 5 ^ { 15 - k } } { 9 ^ { k } }$ 
D)  $\sum _ { k = 1 } ^ { 8 } \frac { 5 ^ { 15 - k } } { 9 ^ { 3 + k } }$

Question 17

Solve the problem.
-A pendulum bob swings through an arc 60 inches long on its first swing. Each swing thereafter, it swings only 90% as far as on the previous swing. How far will it swing altogether before coming to a complete stop?

Accepted Answer

A)  67 inches 
B)  133 inches 
C)  300 inches 
D)  600 inches 
A)  67 inches 
B)  133 inches 
C)  300 inches 
D)  600 inches

Question 18

Determine whether the given sequence is arithmetic, geometric, or neither. If arithmetic, find the common difference. If
geometric, find the common ratio.
-{4n - 2}

Accepted Answer

A)  Arithmetic, d = -2 
B)  Geometric, r = 4 
C)  Neither 
D)  Arithmetic, d = 4 
A)  Arithmetic, d = -2 
B)  Geometric, r = 4 
C)  Neither 
D)  Arithmetic, d = 4

Question 19

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 a9 when $a _ { 1 } = 2 , r = - 2$

Accepted Answer

A)  -14 
B)  512 
C)  -1,024 
D)  516 
A)  -14 
B)  512 
C)  -1,024 
D)  516

Question 20

The given pattern continues. Write down the nth term of the sequence suggested by the pattern.
-$$\frac { 1 } { 1 } , \frac { 1 } { 4 } , \frac { 1 } { 9 } , \frac { 1 } { 16 } , \frac { 1 } { 25 } , \ldots$$

Accepted Answer

A)  $a _ { n } = \frac { 1 } { 3 n - 2 }$ 
B)  $a _ { n } = \left( \frac { 1 } { 2 } \right) ^ { n - 1 }$ 
C)  $a _ { n } = \frac { 1 } { n ^ { n - 1 } }$ 
D)  $a _ { n } = \frac { 1 } { n ^ { 2 } }$ 
A)  $a _ { n } = \frac { 1 } { 3 n - 2 }$ 
B)  $a _ { n } = \left( \frac { 1 } { 2 } \right) ^ { n - 1 }$ 
C)  $a _ { n } = \frac { 1 } { n ^ { n - 1 } }$ 
D)  $a _ { n } = \frac { 1 } { n ^ { 2 } }$

Find the sum of the sequence. - $\sum _ { k = 4 } ^ { 7 } 6 k$

Write out the sum. Do not evaluate. - $\sum _ { k = 1 } ^ { n } 2 ^ { k + 1 }$

Write out the first five terms of the sequence. - $\left\{ 2 ^ { n } \right\}$

Find the indicated coefficient or term. -The 12th term in the expansion of $( x - 2 y ) ^ { 13 }$

Express the sum using summation notation. - $4 ^ { 2 } + 8 ^ { 3 } + 12 ^ { 4 } + \ldots + 32 ^ { 9 }$

Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n. - $\text { Show that the statement "n } { } ^ { 2 } - n + 3 \text { is a prime number" is true for } n = 1 \text {, but is not true for } n = 3 \text {. }$

Solve the problem. -Jennifer takes a job with a starting salary of $40,000 for the first year with an annual increase of 4% beginning in the second year. What is Jennifer's salary, to the nearest dollar, in the sixth year?

Expand the expression using the Binomial Theorem. - $( x + 1 ) ^ { 6 }$

Express the sum using summation notation with a lower limit of summation not necessarily 1 and with k for the index of summation. - $4 + \frac { 9 } { 2 } + 5 + \frac { 11 } { 2 } + \ldots + \frac { 15 } { 2 }$

Determine whether the given sequence is arithmetic, geometric, or neither. If arithmetic, find the common difference. If geometric, find the common ratio. - $\left\{ 3 n ^ { 2 } \right\}$

Find the value of the annuity with the following characteristics. -Payments of $100 are made at the end of each year for 7 years at 4% interest compounded annually

Find the indicated coefficient or term. -The 3rd term in the expansion of ( $( 7 x + 9 ) ^ { 3 }$

Determine whether the infinite geometric series converges or diverges. If it converges, find its sum. - $1 + \frac { 1 } { 2 } + \frac { 1 } { 4 } + \cdots$

Determine whether the infinite geometric series converges or diverges. If it converges, find its sum. - $\sum _ { k = 1 } ^ { \infty } 4 \left( \frac { 2 } { 3 } \right) ^ { k - 1 }$

Express the sum using summation notation. - $\frac { 5 ^ { 14 } } { 9 ^ { 4 } } + \frac { 5 ^ { 13 } } { 9 ^ { 5 } } + \frac { 5 ^ { 12 } } { 9 ^ { 6 } } + \ldots + \frac { 5 ^ { 7 } } { 9 ^ { 11 } }$

Solve the problem. -A pendulum bob swings through an arc 60 inches long on its first swing. Each swing thereafter, it swings only 90% as far as on the previous swing. How far will it swing altogether before coming to a complete stop?

Determine whether the given sequence is arithmetic, geometric, or neither. If arithmetic, find the common difference. If geometric, find the common ratio. -{4n - 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 a9 when $a _ { 1 } = 2 , r = - 2$

The given pattern continues. Write down the nth term of the sequence suggested by the pattern. - $\frac { 1 } { 1 } , \frac { 1 } { 4 } , \frac { 1 } { 9 } , \frac { 1 } { 16 } , \frac { 1 } { 25 } , \ldots$

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

Find the sum of the sequence. - ∑k=476k\sum _ { k = 4 } ^ { 7 } 6 k∑k=47​6k

Write out the sum. Do not evaluate. - ∑k=1n2k+1\sum _ { k = 1 } ^ { n } 2 ^ { k + 1 }∑k=1n​2k+1

Write out the first five terms of the sequence. - {2n}\left\{ 2 ^ { n } \right\}{2n}

Find the indicated coefficient or term. -The 12th term in the expansion of (x−2y)13( x - 2 y ) ^ { 13 }(x−2y)13

Express the sum using summation notation. - 42+83+124+…+3294 ^ { 2 } + 8 ^ { 3 } + 12 ^ { 4 } + \ldots + 32 ^ { 9 }42+83+124+…+329