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!
Answer:
The function for the model that gives the pressure in atm is given as:
P = 1.000 + 0.030*D
Step-by-step explanation:
From the question, we are given the Depth with corresponding P. Our duty is to calculate the Pressure (P) in the atm given the Depth (D). From this, we know that:
1. Pressure (P) is the dependent variable, and
2. Depth (D) is the independent variable.
Hence the function is a linear relationship which is known as regression equation. This can be achieve by estimating the:
i) Slope (β), and
ii) Intercept (α).
We can obtain the slope by:
, and
.
Hence, the function is:
= α + β D
You can achieve the task through R programming. Here is the R code.
#... this code in R programming
D = c(0,33,66,99,132)
P = c(1,2,3,4,5)
mod = lm(P~D)
mod.
############################
And we obtained P = 1.000 + 0.030 D
Explication is 3 minus 2b is 9 because I look it up
I hope this helps you
x= 2 y= -3