Compute 90, 91, 92, 93, 94, and 95. 90 = 1 91 = 9 92 = 81 93 = 729 94 = 6561 95 = 59049 Make a conjecture about the units digit of 9n, where n is a positive integer. If n is even, the units digit of 9n is 1 . If n is odd, the units digit of 9n is 9 . Use strong mathematical induction to prove your conjecture. Submit your proof as a free response. (Submit a file with a maximum size of 1 MB.)

Answers

Answer 1

Answer:

Step-by-step explanation:

First we compute [tex]9^n[/tex], for [tex]n = 0,1,2,3,4,5[/tex]

[tex]9^0=1,9^1=9,9^2=81,9^3=729,9^4=6561,9^5=59049[/tex]

We have the unit digit of [tex]9^n[/tex], is either 1 or 9

Therefore, the conjecture can be state as follows

Conjecture; the unit of digit of [tex]9^n[/tex], is 1, when n is even and 9 when n is odd

Let the property p(n) be the formula;

"The unit digit of [tex]9^n[/tex], is 1 when n is even and 9 when n is odd"↔ P(n)

To show p(n) is true , we will use strong mathematical inductive

Show tgat p(0) and p(1) are true

We have  to show the unit digit of [tex]9^0[/tex] is 1 and [tex]9^1[/tex] is 9

Since any integer with zero power is one and hence

[tex]9^0=1[/tex] and [tex]9^1 = 9[/tex]

Therefore, p(0) and p(1) are true

Show that for al integer [tex]k\geq 1[/tex], if  p(i) is true for all integer i from o through k, then p(k+1) is also true

let k be any integer with [tex]k \geq 1[/tex] and suppose that for all integer i with [tex]0 \leq i \leq \leq k[/tex]

the unit digit of [tex]9^i[/tex] is 1 when is even and 9 when n is odd

we must show that [tex]9^{k +1}[/tex] equals 1 when n is even and 9 when n is odd

Case I; (k +1 is odd)

in this case k is even and so, by inductive hypothesis, the unit digit of [tex]9^k[/tex] is 1 and hence there is some non negative integer q such that

[tex]9^k = 10q + 1[/tex]

now, [tex]9^{k+1}=9^k(9)[/tex]

= (10q + 1)9

= 90q + 9

= 10.9q + 9

Note that for any non negative integer q , 9q is also an integer and this implies the unit digit of [tex]9^{k+1}[/tex] is 9

Case II

(k +1 is even)

in this case k is odd and so, by inductive hypothesis, the unit digit of [tex]9^k[/tex] is 9 and hence there is some non negative integer q such that

[tex]9^k=10q+9[/tex]

Now,

[tex]9^{k+1}=9^k.9\\=(10q+9)9\\=90q+81\\=10(9q+8)+1[/tex]

Note that any non negative integer q, 9q + 8 is also an integer and this implies the unit of [tex]9^{k+1}[/tex] is 1

Since we have proved both the basis and the inductive step of the strong mathematical induction, we conclude that the given statement is true


Related Questions

y varies directly as x.y = 52 when x=4. Find y when x= 10.
y=
(Simplify your answer.)

Answers

Answer:

9ha

shhsi a sa.anjwkw a zkaq

Daniel has $5000 in a savings account, has $7000 in investments, owns a
$10,000 car, and owns a $32,000 house. What is the total value of his large
assets?
O A. $42,000
B. $22,000
C. $12,000
D. $54,000

Answers

Answer:$42,000

Step-by-step explanation:

The total value of his large assets is $42,000.

What is large assest value?

Asset size is the total market value of the securities in a fund. It can also be referred to as assets under management.

Daniel has $5000 in a savings account, has $7000 in investments, owns a $10,000 car, and owns a $32,000 house.

The total value of his large assets is determined in the following steps given below.

[tex]\rm Large \ assest = House + Car\\\\ Large \ assest = 32000+10000\\\\ Large \ assest = 42000[/tex]

Hence, the total value of his large assets is $42,000.

Learn more about assets value here;

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Every year, census researchers collect data from 850 public and private hospitals across the United States. Which issue will invalidate the conclusion of the hypothesis test? Group of answer choices The data comes from only 850 hospitals in the U.S. when there are 5,700 hospitals in the U.S. Researchers received more data from public hospitals than from private hospitals. The data is selected by choosing the first 850 hospitals from an alphabetized list.

Answers

Answer:

Option C

Step-by-step explanation:

For a better, clear and unbiased sampling, choosing randomly might not allow for a biased sampling. Choosing in an alphabetical order might not give a representative of the whole hospitals both private and public, thus invigilator the conclusion of our hypothesis test.

£48 is divided between Alf and Ted in the ratio of 7:9. How much does Alf receive?

Answers

So basically you add up the ratio and get 16 because 7+9.
Now you do 48 divided by 16 which is 3 to get one price. Since Alf is divided by 7 you multiply 3 with 7. So the answer would be £21

7:9=7+9=16

Alf is 7 which means 7/16*£48=6.125

Consider purchasing a system of audio components consisting of a receiver, a pair of speakers, and a CD player. Let A1 be the event that the receiver functions properly throughout the warranty period, A2 be the event that the speakers function properly throughout the warranty period, and A3 be the event that the CD player functions properly throughout the warranty period. Suppose that these events are (mutually) independent with P(A1)= .95, P(A2) = .98, and P(A3) = .80.a. What is the probability that all three components function properly throughout the warranty period?b. What is the probability that at least one component needs service during the warranty period?c. What is the probability that all three components need service during the warranty period?

Answers

Answer:

that is alot the prob. is 20

Step-by-step explanation:

(-3x + 15) + (-3x + 2)

Answers

Answer:

If you are looking for a simplified equation, then the answer is -6x + 17  

Step-by-step explanation:

(-3x + 15) + (-3x + 2)

Because you are only adding and subtracting here, you can take out the parentheses.

-3x + 15 -3x + 2

= -6x + 17  

After flooding damaged much of the crop, the cost of a head of lettuce jumped from $0.92 to $2.26. What percent of increase is this? Round to the nearest one percent.

Answers

Answer:

146%

Step-by-step explanation:

[tex] percent~change = \dfrac{new~price - old~price}{old~price} \times 100% [/tex]

If the percent change is positive, it is a percent increase.

If the percent change is negative, it is a percent decrease.

Your numbers are

old price = $0.92

new price = $2.26

Plug in the numbers in the formula above and evaluate the expression.

percent change = ($2.26 - $0.92)/($0.92) * 100%

percent change = ($1.34)/($0.92) * 100%

percent change = 1.4565 * 100%

percent change = 146%

Since the percent change is a positive number, it is a percent increase.

Answer: The percent increase is 146%.

subtract 6066 from 294

Answers

Answer:

-5772

Step-by-step explanation:

294-6066=-5772

Answer:

The answer would be 2,772

Step-by-step explanation:

6,066 - 294 = 2,272

I hope that helps!!:)

A sculpture sits on a pedestal in the shape of a square prism. The side lengths of a base of the prism are 3 feet. The height of the prism is 4 feet. The museum director wants to cover all but the underside of the pedestal with foil that costs $0.22 per square foot. How much will the foil cost?

Answers

Answer:

Total cost of covering the pedestal is $12.54.

Step-by-step explanation:

The pedestal is a square prism so top and bottom are of square shape.

Hence, we have here length = width

We know that the Total Surface Area of a Prism is:

[tex]\text{Total Surface Area = } 2 (l\times w+ w \times h+ h \times l)[/tex]

Where l is length,

w is width and

h is height of prism

We know that here w = l = 3 feet

and h = 4 feet

Also, underside of pedestal is not to be covered.

The underside of pedestal is square type.

So,

[tex]\text{Required Total Surface Area = } l\times w+ 2\times w \times h+2\times h \times l[/tex]

[tex]\Rightarrow 3 \times 3 + 2 \times 3 \times 4 + 2 \times 3 \times 4\\\Rightarrow 9 + 24 + 24\\\Rightarrow 57\ ft^{2}[/tex]

Cost of cover per square foot = $ 0.22

Total cost of [tex]57\ ft^{2}[/tex]  [tex]\Rightarrow 0.22 \times 0.57 = \$12.54[/tex]

Today a shoe store it’s at 20% of the price of a pair shoes and for the next three days it will take 20% of the previous days price if the price of a pair of shoes yesterday was $200 what will be the price of the pair of shoes three days from now
A.160.00
B.128.00
C.102.40
D.81.92

Answers

If the price of the shoe yesterday was $200 and on this day 20 percent has been taken thus it will have $160. In the next 3 days if another 20 percent will be taken out from the previous $160 then the final answer is $128.

ps.. can u give brainly pls :3

A store display shows two red shirts, one blue shirt, and three shirts with red and white stripes. The display also shows two pairs of blue jeans, one pair of white pants, and one pair of white shorts. What is the probability of randomly selecting an item with white or red on it?
A) 1/4 B) 3/10 C)1/2 D) 7/10

Answers

Answer:

its 7/10

Step-by-step explanation:

The probability of randomly selecting an item with white or red on it is 3/10.

What is the probability?

Probability is a measure of the likelihood of an event to occur. Many events cannot be predicted with total certainty.

A store display shows two red shirts, one blue shirt, and three shirts with red and white stripes.

The display also shows two pairs of blue jeans, one pair of white pants, and one pair of white shorts.

The probability of randomly selecting an item with white or red on it is given by;

[tex]\rm Probability=\dfrac{Total \ number \ of \ items}{Total \ red \ or \ white \ items}\\[/tex]

Total items =  2 + 1 + 3 + 2 + 1 + 1 = 10 itens.

Substitute all the values in the formula

[tex]\rm Probability=\dfrac{Total \ number \ of \ items}{Total \ red \ or \ white \ items}\\\\\rm Probability=\dfrac{3}{10}\\\\[/tex]

Hence, the probability of randomly selecting an item with white or red on it is 3/10.

Learn more about probability here;

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Giving brainliest for CORRECT awnser.

Answers

Answer:

B.

Step-by-step explanation:

So, first off, the quadratic formula is -b plus or minus sqrt(b^2-4ac)/2a. Plug the equation given, and you get B.

the price increase of a set of golf clubs is from $250 to $750. What is the percent increase?

Answers

Answer:

300%

Step-by-step explanation:

3 times 250 is 750

A shoe manufacturer was investigating the weights of men's soccer cleats. He felt that the weight of these cleats was less than the average weight of 10 ounces. After a random sample of 13 pairs of cleats, he found that sample mean was 9.63 and the standard deviation was .585. At a significance level of .05, can it be concluded that the researcher was correct?
Show work

Answers

Answer:

 The conclusion is that the researcher was correct

Step-by-step explanation:

From the question we are told that

     The sample size is  [tex]n = 13[/tex]

    The sample mean is  [tex]\= x = 9.63[/tex]

     The standard deviation is  [tex]s = 0.585[/tex]

      The significance level is  [tex]\alpha = 0.05[/tex]

The Null Hypothesis is  [tex]H_o : \mu = 0[/tex]

The Alternative  Hypothesis  is  [tex]H_a = \mu < 10[/tex]

The test statistic is  mathematically represented as

          [tex]t = \frac{\= x - \mu }{\frac{s}{\sqrt{n} } }[/tex]

Substituting values

          [tex]t = \frac{9.63 - 10 }{\frac{0.585}{\sqrt{13} } }[/tex]

         [tex]t = - 2.280[/tex]

Now the critical value for [tex]\alpha[/tex] is  

     [tex]t_{\alpha } = 1.645[/tex]

This obtained from the critical value table

  So comparing the critical value of alpha and the test value we see that the test value is less than the critical value so the Null Hypothesis is rejected

 The conclusion is that the researcher was correct

 

 

what is the area of a circle with the equation (x − 6)² + (y + 2)² = 17 ?

Answers

Answer:

17π

Step-by-step explanation:

Equation of a circle is:

(x − h)² + (y − k)² = r²

Area of a circle is:

A = πr²

r² = 17, so A = 17π.

A tennis ball machine shoots 21 forehands 25 backhands and 9 lobs if a player missed 6 forehands 17 backhands and 2 lobs how many balls were hit

Answers

Answer:

30

Step-by-step explanation:

55 - 25 = 30

CAN SOME HELP ME PLEASE IT'S DUE IN 10 MINS​

Answers

Answer:

Step-by-step explanation:the first one

Describe how to graph point K at (5, 4) the steps

Answers

Answer:

here is ur answer

Please help ASAP! I will mark Brainliest! Please answer CORRECTLY! No guessing!

Answers

Answer:

Step-by-step explanation:

B or c

Using tan(A+B), find the exact value of tan(105).

Answers

Answer:

  -2-√3

Step-by-step explanation:

The sum of angles formula for tangent is ...

  tan(A+B) = (tan(A) +tan(B))/(1 -tan(A)tan(B))

For the desired angle, we can use A=45°, B=60°. Then we have ...

  tan(105°) = (tan(45°) +tan(60°))/(1 -tan(45°)tan(60°))

The tangent values we need are ...

  tan(45°) = 1

  tan(60°) = √3

Then our tangent is ...

  tan(105°) = (1 +√3)/(1 -1·√3) = (1 +√3)²/((1 -√3)(1 +√3)) = (4+2√3)/(-2)

  tan(105°) = -(2+√3)

Please help ASAP! I will mark Brainliest! Please READ the question THEN answer CORRECTLY! No guessing!

Answers

Answer:

its C

Step-by-step explanation:

Hope this helps!!

Answer:

B. Week 6

Step-by-step explanation:

[tex]f(x) = 2^{x}[/tex]

We know that f(x) represents number of subscribers, x represents week number. So plug in 64 for f(x).

[tex]64 = 2^{x}[/tex]

Uh-oh, how do we solve this? We need to go from exponential form to logarithmic form.

[tex]64 = 2^{x}[/tex]

log₂(64) = x

We're basically asking the question: 2 to the what equals 64? And we can plug this into the calculator as log₂(64) (on TI-84, it's MATH->logBASE(). Or you can just think about it:

2 x 2 x 2 x 2 x 2 x 2 = 64.

There are 6 2s, so 6 is the answer.

Peter rolls a fair dice 114 times. How many times would Peter expect to roll a number greater than 1?

Answers

Answer:

A dice of six faces has 5 numbers bigger than 1 (from 2 to 6).

Then, the number Peter is expecting is : 114 x (5/6) = 95 (times)

Hope this helps!

:)

HELP ME ASAP! Will give BRAINLIEST! Please read the question THEN answer correctly! No guessing. Check ALL that apply

Answers

The possible x-values of the equation are options C and F. These are the + and - values that make the equation true.

what are three segments from this figure?

Answers

Answer:

AD, DA, ED, DE

Step-by-step explanation:

Please help asap! Will give brainliest! Please read the question then answer correctly! No guessing.

Answers

Answer:D

Step-by-step explanation:

(4/5)^0

(4^0)/(5^0)=1/1=1

Find the sum of the interior angle measures of a convex octagon (an eight-sided polygon).

Answers

Answer:

1080

Step-by-step explanation:

:)


f(x)=7x+3 and g(x)=-6x-8, find (f•g)(x) and (f•g)(-1)

Answers

Answer:

[tex](f*g)(x)=-42x^2-74x-24[/tex]

[tex](f*g)(-1)=8[/tex]

Step-by-step explanation:

[tex]f(x)=7x+3\\ g(x)=-6x-8[/tex]

[tex](f*g)(x)=(7x+3)(-6x-8)\\(f*g)(x)=-42x^2-56x-18x-24\\(f*g)(x)=-42x^2-74x-24[/tex]

Now;

[tex](f*g)(-1)=-42x^2-74x-24\\(f*g)(-1)=-42(-1)^2-74(-1)-24\\(f*g)(-1)=-42(1)+74-24\\(f*g)(-1)=-42+50\\(f*g)(-1)=8[/tex]

Alice and Bob share some money in the
ratio 7:5. Alice got £4 more than Bob. How
much did Alice get?

Answers

Answer:

14

Step-by-step explanation:

7-5 = 2

4/2 = 2

2 x 7 = 14

Jonas is buying a three-day supply of camping snacks. An energy bar costs $1.50, and a bag of dried
mangoes costs $3.00. Jonas plans to have 7 snacks per day and can spend up to $45.00.
If p is the number of energy bars he can buy, and m is the number of bags of mangoes he can buy,
which system of equations and inequalities can be used to represent the possible number of each
snack he can buy?
p+m
21
=
21
p+m = 45
1.5p + 3m
p> 0
m >0
1.5p + 3m < 45
p> 0
m 2 0
1.5p + 3m
3p + 1.5m
p20
m 20
p+m = 21
8p + 1.5m
p> 0
m >0

Answers

Answer:

P+m=21

1.5p + 3m < 45

p > 0

m > 0

Step-by-step explanation:

What is 4/12 x 70.please help me

Answers

Answer:

70/3

Step-by-step explanation:

Your welcome :)

Step-by-step explanation:

4 / 12 = 1/3 = 0.33

0.33 x 70 = 23. 33

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