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 is the ratio of 18 inches to 1 yard? (Change 1 yard to 36 inches before writing the ratio)
Answer:
1:2
Step-by-step explanation:
18 inches to 1 yard
We need to convert 1 yard to 36 inches
18 inches : 36 inches
Divide each side by 18
18/18 : 36/18
1:2
5th grade math. correct answer will be marked brainliest, answer the question please!
I put the work and answer in the picture attached!
For some reason Brainly thought I had swear words in here so I took a picture of my answer instead. :)
Aubrey was buying candy for goodie bags for her birthday party. She bought 6 pounds of Jelly beans. How much did she spend on Jelly Beans? (please don´t guess)
Answer:
$8.35
Step-by-step explanation:
6 pounds is $8.35
You and 3 friends meet at Magnolia Bakery. Your bill is $76.20 & you leave a 15% tip. What is the total bill?
Answer:
$87.63
Step-by-step explanation:
Given data
Your bill= $76.20
Tip= 15%
Let us find the amount of 15% tip
=15/100*76.20
=0.15*76.20
=$11.43
Hence, the total bill is
=11.43+76.20
=$87.63
Write the quadratic equation whose roots are 6 and 2, and whose leading coefficient is 2
Answer:
Step-by-step explanation:
y = 2(x-6)(x-2) = 2(x²-8x+12) = 2x² - 16x + 24
which pair of terms has 4a as the greatest common factor ? 8a and 18a , 8b and 16a , 12a and 16ab , 20a and 27a
Answer:
12a and 16ab
Step-by-step explanation:
What is the right answer
Answer:
Earth, Sun, Sun, Earth
Step-by-step explanation:
I have $460 less than dotun. Altogether we have $1280. How much do we have
Answer:
you have $180, total there is $1280
Step-by-step explanation:
The required amount we have is given as $410, as of the given condition.
What are equation models?The equation model is defined as the model of the given situation in the form of an equation using variables and constants.
here,
Let the amount we and Dotun has to be x and y,
According to the question,
x = y - 460- - -- - -(1)
x + y = 1280 - - - (2)
From equation 1
y - 460 + y = 1280
2y = 1280 + 460
y = 1740/2
y = $870
Now,
x = 870 - 460
x = $410
Thus, the required amount we have is given as $410, as of the given condition.
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pls help me im sooo stuck on math
In a cafeteria, 1/6 of the students are eating salads , and 2/3 are eating sandwiches. There are 12 students in the cafeteria. How many students are eating lunches other than salads or sandwiches?
Answer:
2 other people
Step-by-step explanation:
NEED HELP ASAP BRAINLIEST TO THE FASTEST.
Yolanda has two grandfather clocks in her home. One must be wound every six days. Other must be wound every 14 days. Yolanda winds up both clocks on December 15. When is the next time she will have to wind both of the clocks on the same day?
Answer:
January 26 or in 42 days
Step-by-step explanation:
The smallest amount of days that both grandfather clocks share is 42 days. 42 days from December 15 is January 26.
hope this helps :)
Question 7 using substitution if you don’t mind. Help please.
y= -2 and x= 1 according to the solution in the picture
Answer:
x = 1
y = -2
Step-by-step explanation:
3x - 2y = 7
4x + y = 2 then y = 2 - 4x
substitute for y:
3x - 2(2 - 4x) = 7
reduce:
3x - 4 + 8x = 7
11x = 11
x = 1
y = -2
I really need help with this!
Answer:
3(2 + 9)
Step-by-step explanation:
The numbers 6 and 27 can be written as factored as
6 = 2 * 3
27 = 3 * 3 * 3
The both have one factor of 3 in common.
The greatest common factor (the GCF) of 6 and 27 is 3.
Now we factor out 3. To do that, we divide both 6 and 27 by 3.
6/3 = 2
27/3 = 9
We see that
6 + 27 = 3 * 2 + 3 * 9,
and the common factor of 3 is in both multiplications.
Finally, we factor out the 3, leaving the results of the divisions inside the parentheses.
6 + 27 = 3 * 2 + 3 * 9 = 3(2 + 9)
Answer: 3(2 + 9)
Five adult tickets and three child tickets for a movie cost £58.
Two adult tickets and eight child tickets for a movie cost £47.
Find the cost of each type of ticket.
Answer:
Step-by-step explanation:
Answer:
so for the first one the cost for all the tickets is 412
for the last one the cost is 333
Step-by-step explanation:
Ms. Wolf goes shopping at Target and spends $38.75. Sales tax is 8%.
Enter the answer for question B: How much does she pay in total?
Answer:
you have to do the sales tax, times the price $3.10
then you add the $3.10 to the original price which is $41.85
Step-by-step explanation:
Help Please!!!
Thanks
Answer:
d i hope please be d please please
Please answer this correctly without making mistakes I want ace expert and genius people to answer this correctly without making mistakes
Answer:
yz + z = -22
Step-by-step explanation:
Ok so you are already the number that y = 10 and z = -2. You will plug these number in the equation in order to find the answer. :-
y = 10 and z = -2
yz + z
10(-2) - 2
-20 - 2
-22
Hope this helps, thank you :) !!
Question 4 of 10
In each family of functions, the
the family
function is the most basic function in
O A simple
O 8. parent
O c. quadratic
O D linear
Answer:
In each family functions, the parent function is the most basic function in the family.
In each family of functions, the parent function is the most basic function
How to complete the blank?A linear function is represented as:
y = mx + b
A quadratic function is represented as:
y = ax^2 + bx + c
The above and others of similar formats are the most basic of their families, and they are referred to as the parent function
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Find the measure of arc JMK.
HELP PLZ DUE IN 10 MIN PLZ I WILL GIVE 200 point if right PLZ
Answer:
(3,1)
Step-by-step explanation:
Answer:
3,1
Step-by-step explanation:
The price of a car was Rs.1,50,000 last year. It has decreased by 10% this
year. The new price is
Answer:
$135000
Step-by-step explanation:
150000*(1-0.1)=150000*0.9=135000
Ahh help!!
The paper water cups near the water jug are cones of radius 2 inches and height 4 inches if a water bottle of cylindrical shape has a radius of 2 inches and a height of 8 inches how many full paper cups of water are required to fill one water bottle?
Answer:
I think its 678.24
Hope it helps!
Divide the given polynomial by the given monomial.
Answer:
Here is the answer. Solved
Some help pls shhsnsshnasnnansnsnsnsnsnsnsnsnsnsnsnsnnsnsnsnsndndndndndndnnrnrnrnrnfn
Answer:
Step-by-step explanation:
B
the answer is A i wish that helped :)
This pyramid has a square base.
The length of each side of the square is 11 cm.
Each triangular face has an area of 62 cm2.
What is the total surface area of the pyramid?
11 cm
Answer:
In a half-time skills contest Addison received $5 for each free throw she made and $10 for each 3- point shot. She earned a total of $140. The amount of free throws she made was four less than twice the amount of three pointers. How many of each did she make?
Answer:
o.o
Step-by-step explanation:
Prove that there is no complex number such that | z | −z = i
The proof is shown below:
What is complex number?A complexb number is the of a real number and an imaginary number. A complex number is of the form a + ib and is usually represented by z. Here both a and b are real numbers. The value 'a' is called the real part which is denoted by Re(z), and 'b' is called the imaginary part Im(z). Also, ib is called an imaginary number.
The alphabet i is referred to as the iota and is helpful to represent the imaginary part of the complex number. Further the iota(i) is very helpful to find the square root of negative numbers.
We know
z= a +ib
and |z|= √a² + b²
So, |z|-z =i
√a² + b² - a- ib=i
Now, comparing the coefficients
b=-1
√a² + b² - a = 0
√a² + (-1)² - a=0
√a² + 1 - a = 0
√a² + 1 = a
a² + 1 = a²
1= 0
Which is not possible.
Hence, there is no complex number such that | z | −z = i
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Solve this system of equations:
y = x - 4
y = 6x - 9
Show your work in the text box below.
also Write your answer as an ordered pair.
Answer:
y=-3 x=1 (1,-3)
Step-by-step explanation:
y=x-4
y=6x-9
y+4=x
y+9=6x
difference = 5
x+5=6x
x=1
y=-3
1,-3
A manufacturer of banana chips would like to know whether its bag filling machine works correctly at the 402 gram setting. It is believed that the machine is underfilling the bags. A 23 bag sample had a mean of 400 grams with a variance of 900. A level of significance of 0.025 will be used. Assume the population distribution is approximately normal. Determine the decision rule for rejecting the null hypothesis. Round your answer to three decimal places.
Answer:
Kindly check explanation
Step-by-step explanation:
H0 : μ ≥ 402
H1 : μ < 402
Test statistic :
n = 23
Sample Mean, x = 400
Variance = 900
σ = sqrt(900) = 30
Test statistic = (x - μ) ÷ σ/sqrt(n)
Test statistic = (400 - 402) ÷ 30/sqrt(23)
Test statistic = - 2 / 6.2554324
Test statistic = - 0.3197221
Test statistic = - 0.3197
Critical value :
Using the Tcritical value calculator
Tcrit; α = 0.025, df = n - 1 = 23 - 1 = 22
Tcritical = 2.074
Reject Null : if Test statistic ≤ Tcritical (left tail test)
Since ;
Test statistic ≤ Tcritical ; We reject the Null
Answer all of these please
1.The product of a number and seven more than the number
2.The product of a number and seven less than the number
3.Eight less than twice a number is fourteen.
4.One less than three times a number is seven.
5. Four more than five times a number is two less than the number.
6.Ten less than a number is three more than six times the number.
7.Twice the sum of a number and 3 is 20.
Answer: 3. 22-8 / 4. 2x3.5 / 5.
Step-by-step explanation:
Step-by-step explanation:
1. 1×8=8
2. 9×2=18
3. 8+14=22÷2=11
4. 2.7
5. -1.5
6. closest -2.5
7.÷2=10-3=7
