Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n.1 ? 2 + 2 ? 3 + 3 ? 4 + . . . + n(n + 1) = 
What will be an ideal response?
First we show that the statement is true when n = 1.
For n = 1, we get 2 = 
= 2.
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 k + 1. That is, we need to show that
So we assume that 
is true and add the next term, 
to both sides of the equation.
1 ? 2 + 2 ? 3 + 3 ? 4 + . . . + k(k + 1) + (k + 1)(k + 2) = 
+ (k + 1)(k + 2)
= 
+ 
= 
= 
Condition II is satisfied. As a result, the statement is true for all natural numbers n.
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Provide an appropriate response.Express 80 inches in standard notation using feet and inches.
A. 6 ft 10 in. B. 2 yd 8 in. C. 2 yd 8 ft D. 1 yd 3 ft 8 in.
Solve the problem.Given that f(x) = ex - 1 + 3, find f-1(x) and give the domain and range of f-1(x).
A. f-1(x) = ln(x - 1) + 3, domain = (3, ?), range = (-?, ?) B. f-1(x) = ln(x - 1) + 3, domain = (-?, ?), range = (-?, ?) C. f-1(x) = ln(x - 3) + 1, domain = (0, ?), range = (0, ?) D. f-1(x) = ln(x - 3) + 1, domain = (3, ?), range = (-?, ?)
Provide an appropriate response.Simplify: 
A. 
B. - 
C. 
D. 
Expand the expression using the Binomial Theorem.(2x - 1)4
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