Answer::
Step-by-step explanation:
About Prove each of the following statements using mathematical induction. (a) Prove that for any positive integer n, 4 evenly divides 32n-1. (b) Prove that for any positive integer n, 6 evenly divides 7n - 1. (c) Prove that for any positive integer n, 4 evenly divides 11n - 7n. (d) Prove that for any positive integer n, 7 evenly divides 9n - 2n.
Answer:
(a) [tex]3^{2{k+1}} - 1 = 4(9k + 2)[/tex]
(b) [tex]7^{k+1} - 1 = 6(7k + 1)[/tex]
(c) [tex]11(k + 1) - 7(k + 1) = 4(k + 1)[/tex]
(d) [tex]9(k+1) - 2(k+1) = 7(k+1)[/tex]
Step-by-step explanation:
Solving (a): For integer [tex]n[/tex], 4 divides [tex]3^{2n} - 1[/tex]
The proof is as follows:
For n = k, we have:
[tex]\frac{3^{2k}- 1}{4} = k[/tex]
Multiply through by 4
[tex]3^{2k} - 1 = 4k[/tex]
[tex]9^k - 1 = 4k[/tex]
Next, prove the statement is true for [tex]n = k+1[/tex]
[tex]3^{2{k+1}} - 1[/tex]
[tex]3^{2{k+1}} - 1 = 9^{{k+1}} - 1[/tex]
[tex]3^{2{k+1}} - 1 = 9^{k} * 9^1 - 1[/tex]
[tex]3^{2{k+1}} - 1 = 9^k * 9 - 1[/tex]
Express -1 as - 9 + 8
[tex]3^{2{k+1}} - 1 = 9^k * 9 - 9 + 8[/tex]
Factorize:
[tex]3^{2{k+1}} - 1 = 9(9^k - 1) + 8[/tex]
Recall that: [tex]9^k - 1 = 4k[/tex]
So, we have:
[tex]3^{2{k+1}} - 1 = 9*4k + 8[/tex]
Factorize
[tex]3^{2{k+1}} - 1 = 4(9k + 2)[/tex]
Since the above mathematical statement is true, then the given statement has been proved
Solving (b): For integer [tex]n[/tex], 6 divides [tex]7^n - 1[/tex]
The proof is as follows:
For n = k, we have:
[tex]\frac{7^k - 1}{6} = k[/tex]
Multiply through by 6
[tex]7^k - 1 = 6k[/tex]
Next, prove the statement is true for [tex]n = k+1[/tex]
[tex]7^{k+1} - 1[/tex]
[tex]7^{k+1} - 1 = 7^k * 7^1 - 1[/tex]
[tex]7^{k+1} - 1 = 7^k * 7 - 1[/tex]
Express -1 as - 7 + 6
[tex]7^{k+1} - 1 = 7^k * 7 - 7 + 6[/tex]
Factorize:
[tex]7^{k+1} - 1 = 7(7^k - 1) + 6[/tex]
Recall that: [tex]7^k - 1 = 6k[/tex]
So, we have:
[tex]7^{k+1} - 1 = 7(6k) + 6[/tex]
Factorize
[tex]7^{k+1} - 1 = 6(7k + 1)[/tex]
Since the above mathematical statement is true, then the given statement has been proved
Solving (c): For integer [tex]n[/tex], 4 divides [tex]11n - 7n[/tex]
The proof is as follows:
For n = k, we have:
[tex]\frac{11k - 7k}{4} = k[/tex]
Multiply through by 4
[tex]11k - 7k = 4k[/tex]
Next, prove the statement is true for [tex]n = k+1[/tex]
[tex]11(k + 1) - 7(k + 1)[/tex]
[tex]11(k + 1) - 7(k + 1) = 11k + 11 - 7k - 7[/tex]
Collect Like Terms
[tex]11(k + 1) - 7(k + 1) = 11k - 7k + 11 - 7[/tex]
[tex]11(k + 1) - 7(k + 1) = 11k - 7k + 4[/tex]
Recall that: [tex]11k - 7k = 4k[/tex]
So, we have:
[tex]11(k + 1) - 7(k + 1) = 4k + 4[/tex]
Factorize
[tex]11(k + 1) - 7(k + 1) = 4(k + 1)[/tex]
Since the above mathematical statement is true, then the given statement has been proved
Solving (d): For integer [tex]n[/tex], 7 divides [tex]9n - 2n[/tex]
The proof is as follows:
For n = k, we have:
[tex]\frac{9k - 2k}{7} = k[/tex]
Multiply through by 7
[tex]9k - 2k = 7k[/tex]
Next, prove the statement is true for [tex]n = k+1[/tex]
[tex]9(k+1) - 2(k+1)[/tex]
[tex]9(k+1) - 2(k+1) = 9k+9 - 2k-2[/tex]
Collect Like Terms
[tex]9(k+1) - 2(k+1) = 9k- 2k+9 -2[/tex]
[tex]9(k+1) - 2(k+1) = 9k- 2k+7[/tex]
Recall that: [tex]9k - 2k = 7k[/tex]
So, we have:
[tex]9(k+1) - 2(k+1) = 7k+7[/tex]
Factorize
[tex]9(k+1) - 2(k+1) = 7(k+1)[/tex]
Since the above mathematical statement is true, then the given statement has been proved
What are two absolute value expressions that are equivalent to |7+9|.
You did not give is choices to choose from.
|7 + 9| is the same as |-7 - 9|.
|16| is the we same as |-16|.
In a bowl of fruit, there are green grapes and black grapes in the ratio 3: 4
If there are 24 green grapes, how many black grapes are there?
Whats 9387c x 732l +99 *DO NOT REPORT TY!*
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do not come until wednesday
Id: 334 911 1797
pass: hey
Answer:
You did not even start the meeting them you want us to join . Silly
Step-by-step explanation:
6871383
how you doing?
how your day going?
Answer:
6871383
Step-by-step explanation:
why dont come until wednsday??
g A construction company has four trucks to carry sand to construction sites. The amount of sand that one truck can transport in one day can be modeled by an exponential distribution with mean of 4 tons for each of the four trucks. The trucks operate independently Find the probability that exactly two out of the four trucks transport more than 4 tons on a given day. What is the median load
PLEASE SOMEONE ANSWER A REAL ANSWER WILL GIVE BRAINLIST
Answer:
a I think please be right
Provide on equivalent ratio. Show your work....
Write the ratio as a unit rate
per HELP PLEASE
Answer:
VHV
Step-by-step explanation:
wheres the question even?
(Urgent) Worth 10 points 7.3
Answer:
[tex] 2228 \: {ft}^{2} [/tex]
Step-by-step explanation:
Required flooring cover
= Area of the school playground
= Area of square with side 40 ft + Area of semicircle with radius (40/2) i.e. 20 ft
[tex] = {(40)}^{2} + \frac{1}{2} \times 3.14 \times {(20)}^{2} \\ \\ = 1600 + \frac{1}{2} \times 3.14 \times 400 \\ \\ = 1600 + 3.14 \times 200 \\ \\ = 1600 + 628 \\ \\ = 2228 \: {ft}^{2} [/tex]
what is bigger 1 yard or 35 inches
Answer:
Step-by-step explanation:
1 yard has 3 feet
1 foot has 12 inches
3 feet have 3 * 12 = 36
36 is greater than 35
1 yard is greater than 35 inches.
PLS HELP TEXT ILL GIVE BRAINLIEST
Answer:
a = -15
Step-by-step explanation:
multiply both sides by 3, then you get a = -15
Answer:
a= -15 for the first one and b= -1 for the 2nd one
Step-by-step explanation:
Find the solution to -1/3y = -9
Answer:
y=27
Step-by-step explanation:
Hey There!
So the whole objective is to get y on one side and the constant on the other
to do so we use inverse operations
-1/3y=-9
divide each side by 1/3
-9/-1/3=-27
we're left with y=-27
Which best describes the triangle shown?
Answer:
a I think please be right
The system of equations y=- x +5 and y= x - 1 is graphed. What is the solution to the system of equations?
Answer:
2(x+1)=3(x-1). 2x+4 = $*-6. -*=-8 .... x=y x=17. 1. x x. X. = 8. X-9=-3x-5 f). 4x=4 x = 1 ... Graphing Linear Equations. Graph ... Solve the inequality and graph the solution on a number line.
the slide at the playground has a height of 6 feet. the base of the slide measured on the ground is 8 feet. what is the length of the slide?
Answer: 10 feet
Step-by-step explanation:
Barb drew a line through the origin and point
(2, 3). Jamal drew a line through the origin
and point (4,6). How can you tell that Barb
and Jamal drew the same line?
What number should go in the space?
Multiplying by 0.25 is the same as decreasing by
%.
Answer:
75% hope this helps, this can be proven by using 100 as a tester
Answer:
75%
Step-by-step explanation:
(number) x 0.25 = 0.25(number)
0.25(number) = 25% (number)
(number) - 75% = 25% (number)
DA is tangent to the circle at A and DC is tangent to the circle at C. Find m
Answer:
D
Step-by-step explanation:
m angle b equals 77 now we have to find the arc of andle b.
so use the equation arc AC = 2(77)
arcAC = 154
360 - 154
=206
206/2
= 103
i have $2 in my pocket what might be in my pocket (number puzzle)
Answer:
$2
Step-by-step explanation:
12(a – 2) = 14 + 3(2b – 1)
5(a + 1) + 2(1 – 2b) = 14
solve with elimantion?
Answer:
(a,b) = (49/9 , 91/18)
Step-by-step explanation:
12a-6b=35
5a-4b=7
multiply both sides
-24+12b= -70
15a-12b=21
elimiate one varibable by adding the equations
-9a= -49
divide both sides
a=49/9
subsitite the value of a
5*49/9-4b=7
solve the equation
b=91/18
a possible solution is
(a,b) = (49/9 , 91/18)
Check it if you want but yeah
6-4x is Less than or equal to 26
What expression is equivalent to 3x - 7 - 6x + 13
Answer:
-3x + 6 and -3(x - 2)
Step-by-step explanation:
Given
[tex]3x - 7 - 6x + 13[/tex]
Required
An equivalent expression
[tex]3x - 7 - 6x + 13[/tex]
Collect Like Terms
[tex]3x - 7 - 6x + 13 = 3x - 6x - 7 + 13[/tex]
[tex]3x - 7 - 6x + 13 = -3x+6[/tex]
This can be simplified further as:
[tex]3x - 7 - 6x + 13 = -3(x-2)[/tex]
To solve the system of equations below, Grace isolated the variable y in the
first equation and then substituted into the second equation. What was the
resulting equation?
2y = 10x
x y²
64 36
= 1
Answer:
D
Step-by-step explanation:
2y=10x
Divide both sides by 2.
y=5x
Now substitute this into the second equation.
Answer:
b. [tex]x^2/64 - 5y^2/36=1[/tex]
Step-by-step explanation: a p e x
2 cups is to 8 cans as 14 cups is to ___ cans?
Answer:
56
Step-by-step explanation:
The ratio is 1:4
14 * 4 = 56
Answer: 56
Step-by-step explanation:
8 ÷ 2 = 4 (cans) - it is 1 cup.
14 × 4 = 56 (cans) - it is 14 cups.
The graph of a function is a line that passes through the coordinates (2, 11) and (8, 14).
Which is an equation in terms of x and y for this function?
y = 12x + 10
y = 23x + 9
y = 32x + 8
y = 2x + 7
Answer:
y=12x+10
Step-by-step explanation:
hope this helps sorry if I'm wrong
The area of a rectangular folder is 63 square inches. The folder is 7 inches wide. How long is
it?
*........* inches
Answer:
9 inches long
Step-by-step explanation: So, this is an area problem. We know that this is a rectangle. This means we use the equasion lw or length times width. We are missing the length so we can divide the area by the width. This is common in problems where you must find a missing number.
Use the applet to select a random sample of 8 students.Repeat to generate many samples. The applet gives the sample proportion for each sample. Examine the variability in the sample proportions you generated with the applet. Which of the following sequences of sample proportions is most likely to occur for 5 random samples of 8 students from this population? Group of answer choices 0.250, 0.125, 0.500, 0.500, 0.875 0.600, 0.600, 0.600, 0.600, 0.600 0.375, 0.625, 0.500, 0.500, 0.875
Answer:
0.250, 0.125, 0.500, 0.500, 0.875
Step-by-step explanation:
Sampling method is used in statistics where the sample is extracted from the population. A random sampling means that there is equal chance of every proportion to be selected in the sample. In the given scenario the sampling method used is random sampling which has 5 random samples of 8 students. There can be sequence for the sampling starting from 0.250.
giving BRAINLIEST for the correct answer !!
Answer:
x<4
Step-by-step explanation:
number 5 please yeah
Answer:
3x<45
Step-by-step explanation:
Simple
Name
Use the given information to answer Problems 7-11.
Leon and Jacey work for a florist. They each program a drone
to deliver boxes of flowers to neighbors. With Leon's program,
he must manually attach the box to the drone. With Jacey's
the drone can pick up one box at a time from a pile.
program,
ram
Please help!!
Answer:
#7: 30 seconds On The Graph The Drone Has Not Moved and The Time Is At 30 Seconds :)
#8 : 20 Seconds On The Graph All The times go Up by 20 So that is the only Possible answer
#9: Leon Drone Is Faster Look At The Graph 60 Seconds = 10 Feet And Jacey's Graph its 80 Seconds For 10 Feet And I could Go On
#10 The Time In Seconds
#11 With slow speed you can guarantee it will not be broken or damaged
I Hope This Helps :) Please Give Brainiest If It Makes Sense To You !
Thanks :)
Your weight on Mars varies
directly with your weight on
Earth. A person weighing 125 lb on
Earth weighs 47.25 lb on Mars,
since Mars has less gravity. If
you weigh 155 lb on Earth, how
much will you weigh on Mars
rounded to the nearest pound?
Answer:
w=58.59
Step-by-step explanation:
47.25/125=w/155
155×47.25/125=155×w/155
7323.75/125=w
w=58.59