An airliner travels 35 miles in 4 minutes. what is its speed in miles per hour?

Use induction to prove: For every integer n > 1, the number n5 - n is a multiple of 5.

nignag [31]

Answer:

we need to prove : for every integer n>1, the number $n^{5}-n$ is a multiple of 5.

1) check divisibility for n=1, $f(1)=(1)^{5}-1=0$ (divisible)

2) Assume that $f(k)$ is divisible by 5, $f(k)=(k)^{5}-k$

3) Induction,

$f(k+1)=(k+1)^{5}-(k+1)$

$=(k^{5}+5k^{4}+10k^{3}+10k^{2}+5k+1)-k-1$

$=k^{5}+5k^{4}+10k^{3}+10k^{2}+4k$

Now, $f(k+1)-f(k)$

$f(k+1)-f(k)=k^{5}+5k^{4}+10k^{3}+10k^{2}+4k-(k^{5}-k)$

$f(k+1)-f(k)=k^{5}+5k^{4}+10k^{3}+10k^{2}+4k-k^{5}+k$

$f(k+1)-f(k)=5k^{4}+10k^{3}+10k^{2}+5k$

Take out the common factor,

$f(k+1)-f(k)=5(k^{4}+2k^{3}+2k^{2}+k)$ (divisible by 5)

add both the sides by f(k)

$f(k+1)=f(k)+5(k^{4}+2k^{3}+2k^{2}+k)$

We have proved that difference between $f(k+1)$ and $f(k)$ is divisible by 5.

so, our assumption in step 2 is correct.

Since $f(k)$ is divisible by 5, then $f(k+1)$ must be divisible by 5 since we are taking the sum of 2 terms that are divisible by 5.

Therefore, for every integer n>1, the number $n^{5}-n$ is a multiple of 5.

3 0

2 years ago

Give at least four different symbols that have been used to represent specific statistical measures. Describe what measure each

natta225 [31]

Answer:

Variance 0²

standard deviation 0×

mean α

median x-bar

Step-by-step explanation:

these are the four different symbols that have been used to represent specific statistical measures like for example, mean which is used to calculate the average value of a sample or population. Variance is the what is expected from the squared standard deviation of a random variable from is average vale/ mean. Median is the value that lies in the middle in a random sample and lastly the standard deviation is the square root of the variance and it measures how the values of sample are spread or how far are they from the mean. these are also commonly use in excel a lot to calculate and evaluate data.

6 0

2 years ago

Talia used 1/3 of a piece of wood for the base of her project 1/4 of the piece of wood for the vertical support and the rest for

Veseljchak [2.6K]

Answer:

The fraction of wood used for horizontal support is $\frac{5}{12}$ .

Step-by-step explanation:

Assume that the total piece of wood is of length 1.

It is provided that Talia used $\frac{1}{3}$ of the piece of wood for the base of he project.

She use $\frac{1}{4}$ of the piece of wood for the for the vertical support.

The remaining wood she used for horizontal support.

The amount of wood used for horizontal support can be computed by the subtracting the amount of wood used for base of the project and vertical support form 1.

Compute the amount of wood used for horizontal support as follows:

$Horizontal\ support=1-Base\ of\ project-Vertical\ support\\= 1 - \frac{1}{3}-\frac{1}{4}\\ =\frac{12-4-3}{12} \\=\frac{5}{12}$

Thus, the fraction of wood used for horizontal support is $\frac{5}{12}$ .

3 0

2 years ago

Tell whether the lines for the pair of equations are parallel, perpendicular, or neither

attashe74 [19]

Answer:

they are parallel to each other

5 0

2 years ago

Jack looks at a clock tower from a distance and determines that the angle of elevation of the top of the tower is 40°. John, who

zmey [24]

Answer:

18.7939 m

Step-by-step explanation:

-Let x be the distance between John and clock tower.

-Let y be the vertical distance from the eyes of the two men standing to the top of the clock tower.

#Taking the right triangle ACD:

$\Tan \ theta=\frac{Perpendicular \ Height}{Base}\\\\Tan \ 60\textdegree=\frac{y+1.5}{x}\\\\y=x \ Tan \ 60\textdegree -1.5$

#Taking the right triangle ABD:

$\Tan \ theta=\frac{Perpendicular \ Height}{Base}\\\\Tan \ 40\textdegree=\frac{y+1.5}{x+20}\\\\y=(x+20)\ Tan \ 40\textdegree -1.5$

#We equate the two yo solve for x and y;

$(x+20)\ Tan \ 40\textdegree -1.5=x\ Tan \ 60\textdegree -1.5\\\\(x+20)\ Tan \ 40\textdegree=x\ Tan \ 60\textdegree\\\\0.8391x+16.7820=1.7321x\\\\x=18.7939$

Hence, John's distance from the tower's base is 18.7939 m

8 0

2 years ago

An airliner travels 35 miles in 4 minutes. what is its speed in miles per​ hour?

An airliner travels 35 miles in 4 minutes. what is its speed in miles per hour?