Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n.1 + 4 + 42 + ... + 4n - 1 = 
What will be an ideal response?
First, we show that the statement is true when n = 1.
For n = 1, we get 1 (or 4[(1) - 1]) = 
= 
= 1.
This is a true statement and Condition I is satisfied.
Next, we assume the statement holds for some k. That is,
is true for some positive integer k.
We need to show that the statement holds for 
. That is, we need to show that
So we assume that 
is true and add the next term, 4k, to both sides of the equation.
Condition II is satisfied. As a result, the statement is true for all natural numbers n.
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Provide an appropriate response.For the students at one college the expected value for "number of siblings" is 1. Which of the following statements is a reasonable conclusion?
A. If a student is selected at random, the most likely number of siblings for the student is 1. B. If a student selected at random has 2 siblings, the next student selected will have no siblings. C. If 4 students are selected at random, the average number of siblings for the 4 students will be 1. D. If 100 students are selected at random, the average number of siblings for the 100 students will be close to 1.
The table shows ice cream cone sales at an ice cream shop. Let p and q represent the following statements:p: Vanilla is the best-selling flavor.q: Chocolate chip outsold strawberry by 13,000 cones. Ice Cream Sold One Year 
Write the symbolic statement below in words. Then use the table to determine the truth value of the statement. ~ q ? ~ p
A. If chocolate chip did not out sell strawberry by 13,000 cones, then vanilla was not the best-selling flavor. Truth value: False B. If chocolate chip out sold strawberry by 13,000 cones, then vanilla was not the best-selling flavor. Truth value: True C. If chocolate chip did not out sell strawberry by 13,000 cones, then vanilla was the best-selling flavor. Truth value: True D. If chocolate chip did not out sell strawberry by 13,000 cones, then vanilla was not the best-selling flavor. Truth value: True
A value of sinh x or cosh x is given. Use the definitions and the identity 
to find the value of the other indicated hyperbolic function.sinh x = 
, coth x =
A. 
B. 
C. 
D. 
Write the English phrase as an algebraic expression. Let the variable x represent the number.Twice a number, increased by 79
A. 2x - 79 B. 2x + 79 C. 2(x + 79) D. x + 79