For (2), start with the base case. When n = 2, we have
(n + 1)! = (2 + 1)! = 3! = 6
2ⁿ = 2² = 4
6 > 4, so the case of n = 2 is true.
Now assume the inequality holds for n = k, so that
(k + 1)! > 2ᵏ
Under this hypothesis, we want to show the inequality holds for n = k + 1. By definition of factorial, we have
((k + 1) + 1)! = (k + 2)! = (k + 2) (k + 1)!
Then by our hypothesis,
(k + 2) (k + 1)! > (k + 2) 2ᵏ = k•2ᵏ + 2ᵏ⁺¹
and k•2ᵏ ≥ 2•2² = 8, so
k•2ᵏ + 2ᵏ⁺¹ ≥ 8 + 2ᵏ⁺¹ > 2ᵏ⁺¹
which proves the claim.
Unfortunately, I can't help you with (3). Sorry!
Let one of those numbers be x.
Now,
according to question,
The other number must be,
3x-6
Now,
as said in the question,
x+3x-6 =62
4x -6 =62
4x = 68
x = 17.
Now,
we got one of those numbers which was x = 17,
and the other number should be 3(17)-6 = 51-6 = 45.
Answer: option C
find f(x) - g(x)
f(x)= 
g(x) = 
To find f(x) - g(x) we plug in the functions and subtract

Remove the parenthesis and change the sign of each term

Combine like terms

So option C is correct
The population function of the Western Lowland Gorillas can either represent population growth or population decay
<h3>How to model the population</h3>
The question is incomplete, as the resources to model the population of the Western Lowland Gorillas are not given.
However, the following is a general guide to solve the question.
An exponential function is represented as:

Where:
- (a) represent the initial value i.e. the initial population of the Western Lowland Gorillas
- (r) represents the rate at which the population increases or decreases.
- (x) represents the number of years since 2022
- (y) represents the population in x years
Given that the population of the Western Lowland Gorillas decreases, then the rate of the function would be 1 -r (i.e. an exponential decay)
Take for instance:

By comparison.
a = 2000 and 1 - r = 0.98
The above function can be used to model the population of the Western Lowland Gorillas
Read more about exponential functions at:
brainly.com/question/26829092
Simplify 7^3 to 343
-343 + 21x^2 + 3x - 9
Collect like terms
(-343 - 9) + 21x^2 + 3x
Simplify
<u>= -352 + 21x^2 + 3x</u>