Question 1

Use mathematical induction to solve for all positive integers n.​ $\sum _ { i = 1 } ^ { n } i ( i + 4 ) = ?$ ​

Accepted Answer

A)  $\frac { n ( 2 n + 13 ) } { 6 }$ 
B)  $\frac { n ( n + 1 ) ( 2 n + 13 ) } { 4 }$ 
C)  $\frac { n ( n + 1 ) ( n + 13 ) } { 6 }$ 
D)  $\frac { n ( n + 1 ) ( 2 n + 13 ) } { 6 }$ 
E)  $\frac { n ( n + 1 ) ( 2 n + 13 ) } { 2 }$ 
A)  $\frac { n ( 2 n + 13 ) } { 6 }$ 
B)  $\frac { n ( n + 1 ) ( 2 n + 13 ) } { 4 }$ 
C)  $\frac { n ( n + 1 ) ( n + 13 ) } { 6 }$ 
D)  $\frac { n ( n + 1 ) ( 2 n + 13 ) } { 6 }$ 
E)  $\frac { n ( n + 1 ) ( 2 n + 13 ) } { 2 }$

Question 2

Determine whether the statement is true or false. ​If the statement P1​ is true but the true statement P6 does not imply that the statement P7 is true,then Pn is not
Necessarily true for all positive integers n.
​
​

Accepted Answer

A) False 
B) True 
A) False 
B) True

Question 3

Find the sum using the formulas for the sums of powers of integers.​ $$\sum _ { n = 1 } ^ { 7 } n ^ { 5 }$$ ​

Accepted Answer

A) 840 
B) 29,008 
C) 4,676 
D) 784 
E) 140 
A) 840 
B) 29,008 
C) 4,676 
D) 784 
E) 140

Question 4

Find the sum using the formulas for the sums of powers of integers.​ $$\sum _ { n = 1 } ^ { 22 } \left( n ^ { 3 } - n \right)$$ ​

Accepted Answer

A) 22,770 
B) 3,795 
C) 63,756 
D) 256,036 
E) 64,009 
A) 22,770 
B) 3,795 
C) 63,756 
D) 256,036 
E) 64,009

Question 5

Use mathematical induction to solve for all positive integers n.​ $1 + 2 + 3 + 4 + \ldots + n = ?$ ​

Accepted Answer

A)  $\frac { n ( n + 1 ) } { 2 }$ 
B)  $\frac { ( n + 1 ) } { 2 }$ 
C)  $\frac { n } { 2 } ( 3 n + 1 )$ 
D)  $\frac { n ( n + 1 ) ( 1 n + 1 ) } { 6 }$ 
E)  $\frac { n ^ { 2 } ( n + 1 ) ^ { 2 } } { 3 }$ 
A)  $\frac { n ( n + 1 ) } { 2 }$ 
B)  $\frac { ( n + 1 ) } { 2 }$ 
C)  $\frac { n } { 2 } ( 3 n + 1 )$ 
D)  $\frac { n ( n + 1 ) ( 1 n + 1 ) } { 6 }$ 
E)  $\frac { n ^ { 2 } ( n + 1 ) ^ { 2 } } { 3 }$

Question 6

Find a quadratic model for the sequence with the indicated terms. ​
A0 = 3,a1 = 3,a4 = 15
​

Accepted Answer

A) an = n2 - n + 15 
B) an = n2 + n - 3 
C) an = n2 - n - 3 
D) an = n2 + n + ​3 
E) an = n2 - n + 3 
A) an = n2 - n + 15 
B) an = n2 + n - 3 
C) an = n2 - n - 3 
D) an = n2 + n + ​3 
E) an = n2 - n + 3

Question 7

Find a formula for the sum of the first n terms of the sequence.​ $9,12,15,18,21 , \ldots$ ​

Accepted Answer

A)  $S _ { n } = \frac { n } { 2 } ( 3 n + 15 )$ 
B)  $S _ { n } = \frac { n } { 2 } ( 3 n + 3 )$ 
C)  $S _ { n } = \frac { n } { 2 } ( - 3 n - 15 )$ 
D)  $S _ { n } = \frac { n } { 2 } ( 3 n - 15 )$ 
E)  $S _ { n } = \frac { n } { 2 } ( - 3 n + 15 )$ 
A)  $S _ { n } = \frac { n } { 2 } ( 3 n + 15 )$ 
B)  $S _ { n } = \frac { n } { 2 } ( 3 n + 3 )$ 
C)  $S _ { n } = \frac { n } { 2 } ( - 3 n - 15 )$ 
D)  $S _ { n } = \frac { n } { 2 } ( 3 n - 15 )$ 
E)  $S _ { n } = \frac { n } { 2 } ( - 3 n + 15 )$

Question 8

Find pk + 1 for the given pk.​ $p _ { k } = \frac { 1 } { 4 ( k + 2 ) }$ ​

Accepted Answer

A)  $p _ { k + 1 } = \frac { 1 } { k ( k + 3 ) }$ 
B)  $p _ { k + 1 } = \frac { 4 } { ( k + 1 ) ( k + 2 ) }$ 
C)  $p _ { k + 1 } = \frac { 4 } { k ( k + 3 ) }$ 
D)  $p _ { k + 1 } = \frac { 1 } { ( k + 3 ) ( k + 2 ) }$ 
E)  $p _ { k + 1 } = \frac { 1 } { 4 ( k + 3 ) }$ 
A)  $p _ { k + 1 } = \frac { 1 } { k ( k + 3 ) }$ 
B)  $p _ { k + 1 } = \frac { 4 } { ( k + 1 ) ( k + 2 ) }$ 
C)  $p _ { k + 1 } = \frac { 4 } { k ( k + 3 ) }$ 
D)  $p _ { k + 1 } = \frac { 1 } { ( k + 3 ) ( k + 2 ) }$ 
E)  $p _ { k + 1 } = \frac { 1 } { 4 ( k + 3 ) }$

Exam 55: Mathematical Induction

Use mathematical induction to solve for all positive integers n.​ ∑i=1ni(i+4)=?\sum _ { i = 1 } ^ { n } i ( i + 4 ) = ?∑i=1n​i(i+4)=? ​

Determine whether the statement is true or false. ​If the statement P1​ is true but the true statement P6 does not imply that the statement P7 is true,then Pn is not Necessarily true for all positive integers n. ​ ​

Find the sum using the formulas for the sums of powers of integers.​ ∑n=17n5\sum _ { n = 1 } ^ { 7 } n ^ { 5 }∑n=17​n5 ​

Find the sum using the formulas for the sums of powers of integers.​ ∑n=122(n3−n)\sum _ { n = 1 } ^ { 22 } \left( n ^ { 3 } - n \right)∑n=122​(n3−n) ​

Use mathematical induction to solve for all positive integers n.​ 1+2+3+4+…+n=?1 + 2 + 3 + 4 + \ldots + n = ?1+2+3+4+…+n=? ​

Find a quadratic model for the sequence with the indicated terms. ​ A0 = 3,a1 = 3,a4 = 15 ​

Find a formula for the sum of the first n terms of the sequence.​ 9,12,15,18,21,…9,12,15,18,21 , \ldots9,12,15,18,21,… ​

Find pk + 1 for the given pk.​ pk=14(k+2)p _ { k } = \frac { 1 } { 4 ( k + 2 ) }pk​=4(k+2)1​ ​

Rectangular Coordinates

Graphs of Equations

Linear Equations in Two Variables

Functions

Analyzing Graphs of Functions

A Library of Parent Functions

Transformations of Functions

Combinations of Functions Composite Functions

Inverse Functions

Mathematical Modeling and Variation

Quadratic Functions and Models

Polynomial Functions of Higher Degree

Polynomial and Synthetic Division

Complex Numbers

Zeros of Polynomial Functions

Rational Functions

Nonlinear Inequalities

Exponential Functions and Their Graphs

Logarithmic Functions and Their Graphs

Properties of Logarithms

Exponential and Logarithmic Equations

Exponential and Logarithmic Models

Radian and Degree Measure

Trigonometric Functions The Unit Circle

Right Triangle Trigonometry

Trigonometric Functions of Any Angle

Graphs of Sine and Cosine Functions

Graphs of Other Trigonometric Functions

Inverse Trigonometric Functions

Applications and Models

Using Fundamental Identities

Verifying Trigonometric Equations

Solving Trigonometric Equations

Sum and Difference Formulas

Multiple Angle and Product to Sum Formulas

Law of Sines

Law of Cosines

Vectors in the Plane

Vectors and Dot Products

Trigonometric Form of a Complex Number

Linear and Nonlinear Systems of Equations

Two Variable Linear Systems

Multivariable Linear Systems

Partial Fractions

Systems of Inequalities

Linear Programming

Matrices and Systems of Equations

Operations With Matrices

The Inverse of a Square Matrix

The Determinant of a Square Matrix

Applications of Matrices and Determinants

Sequences and Series

Arithmetic Sequences and Partial Sums

Geometric Sequences and Series

The Binomial Theorem

Counting Principles

Probability

Lines

Introduction to Conics Parabolas

Ellipses

Hyperbolas

Rotation of Conics

Parametric Equations

Polar Coordinates

Polar Equations of Conics

Graphs of Polar Equations

Filters

Use mathematical induction to solve for all positive integers n. $\sum _ { i = 1 } ^ { n } i ( i + 4 ) = ?$

Determine whether the statement is true or false. If the statement P₁ is true but the true statement P₆ does not imply that the statement P₇ is true,then P_n is not Necessarily true for all positive integers n.

Find the sum using the formulas for the sums of powers of integers. $\sum _ { n = 1 } ^ { 7 } n ^ { 5 }$

Find the sum using the formulas for the sums of powers of integers. $\sum _ { n = 1 } ^ { 22 } \left( n ^ { 3 } - n \right)$

Use mathematical induction to solve for all positive integers n. $1 + 2 + 3 + 4 + \ldots + n = ?$

Find a quadratic model for the sequence with the indicated terms. A₀ = 3,a₁ = 3,a₄ = 15

Find a formula for the sum of the first n terms of the sequence. $9,12,15,18,21 , \ldots$

Find pk_{+ 1} for the given pk. $p _ { k } = \frac { 1 } { 4 ( k + 2 ) }$