1.6.IP-20

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

3 years ago

By switching service providers , a family's telephone bill decreased from about $ 50 a month to about 44 . What was the percent

anyanavicka [17]

Answer:

12% decrease

Step-by-step explanation:

50 - 44 = 6

6/50 = 0.12 or 12% decrease

6 0

3 years ago

What is the approximate area of the shaded portion of the diagram? (Use 3.14 as an estimate for pi.)

Nadusha1986 [10]

Answer:

218

Step-by-step explanation:

first find the area of the circle-

$A=\pi r^{2} \\A=\pi 10^{2} \\A=100\pi \\A=314$

to find the area of the triangle, we need the base and the height. The base we know is 16, since we don't know the height yet, use the Pythagorean theorem to find the height. One leg is 16 the hypotenuse is 20 (2* the radius).

$a^{2} +b^{2} =c^{2} \\a^{2} +16^{2} =20^{2} \\a^{2} +256=400\\a^{2} =144\\a=12$

Now I know the height is 12. Find the area of the triangle

$A=\frac{1}{2} bh\\A=\frac{1}{2} (16)(20)\\A=96$

subtract the area of the triangle from the area of the circle

314-96=218

6 0

3 years ago

Which factor does the width of the peak of a normal curve depend on?

Aleks [24]

<span>The normal curve is described by a bell curved graph. The bell curved graph of a normal distribution depends on two factors: the mean and the standard deviation. The mean identifies the position of the center and the standard deviation determines the height and width of the bell. Therefore, the factor that the width of the peak of the normal curve depends on is the standard deviation.</span>

4 0

3 years ago

Read 2 more answers

What is the answer to this question (3x3)+7-7=

exis [7]

Answer:

9

Step-by-step explanation:

Assuming x is the multiplication symbol, you would multiply 3 and the other 3 together. This leaves you with 9+7-7. Since 7-7=0 you can just get rid of both sevens completely. Now you are only left with 9.

6 0

3 years ago

Read 2 more answers