I think so that the answer is -48828125
Basis step: Let
. Then
![2! = 2\cdot1 = 2](https://tex.z-dn.net/?f=2%21%20%3D%202%5Ccdot1%20%3D%202)
and
![2^2 = 4](https://tex.z-dn.net/?f=2%5E2%20%3D%204)
so
.
Inductive step: Assume
. We want to show that
.
Now
![(k+1)! = (k+1) \cdot k! \\\\ ~~~~~~~~~~~ < (k+1) \cdot k^k \\\\ ~~~~~~~~~~~ < (k+1) \cdot k^{k+1} \\\\ ~~~~~~~~~~~ < (k+1) \cdot (k+1)^{k+1}](https://tex.z-dn.net/?f=%28k%2B1%29%21%20%3D%20%28k%2B1%29%20%5Ccdot%20k%21%20%5C%5C%5C%5C%20~~~~~~~~~~~%20%3C%20%28k%2B1%29%20%5Ccdot%20k%5Ek%20%5C%5C%5C%5C%20~~~~~~~~~~~%20%3C%20%28k%2B1%29%20%5Ccdot%20k%5E%7Bk%2B1%7D%20%5C%5C%5C%5C%20~~~~~~~~~~~%20%3C%20%28k%2B1%29%20%5Ccdot%20%28k%2B1%29%5E%7Bk%2B1%7D)
where the first inequality follows from the induction hypothesis; the second follows from multiplying the right side by some
; and the third from adding the remaining terms to complete the binomial expansion
, all of which are positive since they are some power of
.
QED
64. 80x4=320. 80 is a b average. 320-252=68. 252 is the total of her current grades.
Answer:
![y=1.25x+3.5](https://tex.z-dn.net/?f=y%3D1.25x%2B3.5)
Step-by-step explanation:
Given:
The relation between the number of tickets and total cost for purchasing them.
Let
represent number of tickets and
represent the total cost to purchase them.
Now, let the relation be given by the following function:
, where,
and
are constants.
Let us find
and
using the values from the table.
For,
. So,
![12.25=7m+b\\or\ 7m+b=12.25------------1](https://tex.z-dn.net/?f=12.25%3D7m%2Bb%5C%5Cor%5C%207m%2Bb%3D12.25------------1)
For,
. So,
![16=10m+b\\or\ 10m+b=16------------2](https://tex.z-dn.net/?f=16%3D10m%2Bb%5C%5Cor%5C%2010m%2Bb%3D16------------2)
Subtract equation 1 from equation 2. This gives,
![10m+b-7m-b=16-12.25\\(10m-7m)+(b-b)=3.75\\3m=3.75\\m=\frac{3.75}{3}=1.25](https://tex.z-dn.net/?f=10m%2Bb-7m-b%3D16-12.25%5C%5C%2810m-7m%29%2B%28b-b%29%3D3.75%5C%5C3m%3D3.75%5C%5Cm%3D%5Cfrac%7B3.75%7D%7B3%7D%3D1.25)
Now, plug in
in equation 1 and solve for
.
![7(1.25)+b=12.25\\8.75+b=12.25\\b=12.25-8.75=3.5](https://tex.z-dn.net/?f=7%281.25%29%2Bb%3D12.25%5C%5C8.75%2Bb%3D12.25%5C%5Cb%3D12.25-8.75%3D3.5)
Therefore, the function to represent the relationship between the tickets and total cost is given as:
![y=mx+b\\y=1.25x+3.5](https://tex.z-dn.net/?f=y%3Dmx%2Bb%5C%5Cy%3D1.25x%2B3.5)