A geometric sequence goes from one term to the next by always multiplying or dividing by the constant value except 0. The constant number multiplied (or divided) at each stage of a geometric sequence is called the common ratio (r).
We have:
[tex]a_1=\dfrac{1}{2}\\\\a_2=2\\\\a_3=8\\\\a_4=32\\\vdots[/tex]
Find the common ratio:
[tex]\boxed{r=\dfrac{a_n}{a_{n-1}}}\Rightarrow r=\dfrac{2}{\frac{1}{2}}=\dfrac{8}{2}=\dfrac{32}{8}=...\\\\\boxed{r=4}[/tex]
The formula of the n-th term of a geometric sequence:[tex]\boxed{a_n=a_1\cdot r^{n-1}}[/tex]
Substitute:
[tex]a_1=\dfrac{1}{2},\ r=4,\ n=8\\\\a_8=\dfrac{1}{2}\cdot4^{8-1}=\dfrac{1}{2}\cdot4^7=\dfrac{1}{2}\cdot16384\\\\\huge\boxed{a_8=8192}[/tex]
Method step by step:[tex]a_4=32,\ r=4\\\\a_5=a_4\cdot r\Rightarrow a_5=32\cdot4=128\\\\a_6=a_5\cdot r\Rightarrow a_6=128\cdot4=512\\\\a_7=a_6\cdot r\Rightarrow a_7=512\cdot4=2048\\\\a_8=a_7\cdot r\Rightarrow a_8=2048\cdot4=8192[/tex]
What is the y-value of the solution to the system of equations {3x+4y=42x−4y=6?
The value of y for the system of equation is -1/2 .
What is a System of Equation ?A system of equation is a set of equation that have common solution
The set of equation given is
3x+4y=4
2x−4y=6
To solve the system , Elimination method will be used
Add equation 1 and 2 to eliminate y
5x = 10
x = 2
Substitute the value of x in equation 1 to determine y
3x +4y = 4
6 + 4y = 4
4y = -2
y = -1/2
Therefore the value of y for the system of equation is -1/2 .
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Arc CD is 1/4 of the circumference of a circle. What is the radian measure of the central angle?
T
Step-by-step explanation:
The radians is just the length of the arc, so if you had a unit circle where the radius was 1, the length of that entire circle would be [tex]2\pi[/tex] since the circumference of any circle is [tex]2\pi r[/tex] except r is 1 so the length is just [tex]2\pi[/tex]. So if you take 1/4 of this you get [tex]\frac{2\pi}{4}[/tex] which can be simplified to [tex]\frac{\pi}{2}[/tex]
Answer:
Step-by-step explanation:
Comments
You need to get the radian measure of the circle.
The radian measure of the entire circle is 2 * pi
To get the radian measure of 1/4 of the circle, just divide by 4.
2 pi / 4 = pi/2
The central angle is = to 1/4 of the circumference of the circle.
So the central angle is also pi/2
Answer: pi/2
If you could answer this thank you!
Answer: 55°
Step-by-step explanation:
Hi!
since an angle bisector splits it in half,
110°, being the entire angle divided by 2 = 110/2 = 55°
What is the relationship between a and b?
Answer:
A
Step-by-step explanation:
If LM = 3x + 5, MN = 4x, and LN
=
11x7, select all that are true
a. x=3
b. x=7
c. lm = 14
d. lm = 28
e. ln = 26
f. ln =54
We have that x = 3, LM = 14 and LN = 2. options A, C and E
How to determine the value
We known that the three sides are on a straight line
LM + MN = LN
Let's substitute the values, we have
3x + 5 + 4x = 11x - 7
Collect like terms
3x + 4x - 11x = -7 -5
-4x = -12
x = -12/ -4
x = 3
LM = 3x + 5 = 3(3) + 5 = 14
LN = 11x - 7 = 11(3) -7 = 33 -7 = 26
Thus, we have that x = 3, LM = 14 and LN = 2. options A, C and E
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(score for question 3: ... of 10 points) 3. the table shows the test scores and the sleep averages of several students. test score (%) 88 75 76 92 96 94 83 90 99 65 7788 82 83 94 97 7 6.5 average sleep (h) 6 7.5 8 7 6.5 8 8.5 5 7 8 9 9 9 8.5 8.5 (a) write the least squares regression equation that models the data. let x = the test score and y = average sleep. (b) use the equation to determine the approximate test score of a student who sleeps an average of 8 hours a night. show your work. answer: 1
The linear regression model obtained by fitting the data points is Y = 0.0914X1 - 0.3741.
How to illustrate the regression?x = test score
y = average sleep
Slope = 0.0914
intercept = - 0.3741
The test score of a student that sleeps 8 hours
y = 8
8 = 0.0914X1 - 0.3741
8 + 0.3741 = 0.0914X1
X = 8.3741 / 0.0914
X = 91.62
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A certain drug decays at a rate of 10% per hour. If the initial dose is 700 mg, how much left in a person’s system after 5 hours?
50
---- ×700=350
100
700-350=350
Therefore 350 grams is left in a person's system after 5 hours
f(x)=x(4x+9)(x-2)(2x-9)(x+5) has zeros at x = -5, x = -9/4, x=0, x=2, and x =9/2. what is the sign of f on the interval 0
Answer:
Just look at the other answer
Step-by-step explanation:
Answer:
f is always positive on the interval.
Step-by-step explanation:
Using the Addition Method, solve for x in the following system of linear equations.
2y + x = 4
y – 3x = 2
a.)
x = 1
b.)
x = 2
c.)
x = 8
d.)
x = 0
Answer:
x = 0
Step-by-step explanation:
if 2y + x = 4
we can assume that y = 2
if y - 3x = 2
it means 2 - 0 = 2
therefore x = 0
Line K has a slope of -5. Line J is perpendicular to line K. What is the slope of line K?
If line J is perpendicular of line K, they should cross to make an X. If the slope of line K is -5, Line J would be 5.
HELP MATES, HELP!
Mustafa was asked whether the following equation is an identity:
Answer: Mustafa is incorrect. He made a mistake in step 2.
Step-by-step explanation: Mustafa was certainly correct in step 1 that x(9x+3) + 3(x+3) is 9x^2 + 6x + 9. However when he was factoring, he made a mistake.
(3x + 3)^2 is NOT equal to 9x^2 + 6x + 9. Rather, (3x + 3)^2 is equal to 9x^2 + 18x + 9. If you want to double check this, use FOIL.
The correct factoring of 9x^2 + 6x + 9 is actually 3(3x^2 + 2x + 3). So, Mustafa made a mistake in step 2.
When you rewrite the general Standard form equation in slope-intercept form, y = mx + b, you can see
that the slope of a standard form equation is
m
and the y-intercept is
b
BC-A
The procedure to rewrite the standard form in slope intercept form is as shown below.
How to rewrite an equation in Slope Intercept Form?The standard form of the equation of a line is;
ax + by + c = 0
Slope intercept form is;
y = mx + c
Intercept form is;
x/p + y/q = 1
To Convert ax + by + c = 0 into y = mx + c
1) Move ax and c to the right hand side to get;
by = -ax - c
Divide both sides by b to get;
y = - ax/b - c/b
Comparing to slope intercept form, we can say that;
Slope = -a/b
y-intercept = -c/b
If we are using y = mx + b as slope intercept form, this will also be accurate because we will just switch c with b in the standard form to arrive at same answer.
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Determine whether the following is a statistical question.
How old are you?
PLEASE HELP ME
Answer:
A statistical question is a question that can be answered by collecting data that vary. For example, “How old am I?” is not a statistical question, but “How old are the students in my school? and "how old are you?" are statistical questions.
Step-by-step explanation:
Your friend is helping to raise money for a local charity by participating in a cartwheel-a-thon. Your
friend is 70 inches tall with your friend's arms raised in the air. Your friend is able to complete a
cartwheel in 15 seconds. The charity sponsor supplies a wristband to each participant to assist in
counting the number of cartwheels completed. The wristband is 4 inches from the end of your friend's
arm. Write a model for the height h (in inches) of the wristband as a function of the time (in minutes)
given that the wristband is at the highest point when + = 0.
The height of 70 inches, and location of the wristband as well as the period of 15 seconds gives the function for the height of the wristband as the equation;
h = 31•sin((8•π)•t + π/2) + 35How can the height of the wristband be modelled?The model for the height can be derived from the general form of the sine function as follows;
y = A•sin(B•t - C) + DMaximum point = 70 - 4 = 66 at t = 0
Minimum point = 4
Amplitude, A = (66 - 4) ÷ 2 = 31
D = 4 + 31 = 35
At t = 0, we have;
66 = 31 × sin(B × 0 - C) + 35
31 = 31 × sin(- C)
sin(- C) = 1
C = -π/2
1 minute = 60 seconds
1 second= 1 minute/60
15 seconds = (15/60) minutes
Period = 15 seconds = 15/60 minutes
Period = 2•π/B
Therefore;
15/60 = 2•π/B
1/4 = 2•π/B
B = 2•π/(1/4) = 8•π
y = Height of the function
Let h represent the height of the wristband.
The equation for the height is therefore;
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Can someone help me out in these geometry fill in the blanks? It’s urgent, gotta have answers fast
Answer:
Vertical angles are equal
Step-by-step explanation:
-
[tex]vertical \: angles \: are \: equal[/tex]
[tex]3y = 150[/tex]
The sum of two supplementary angles equals 180 degrees
Therefore 6x +150=180
Figure AAA is a scale image of figure BBB.
Figure AAA maps to figure BBB with a scale factor of \dfrac{2}{7}
7
2
start fraction, 2, divided by, 7, end fraction.
What is the value of xxx?
Figure A maps to figure B with a scale factor of 4/9, hence the value of x is x = 54
What is a transformation?Transformation is the movement of a point from its initial point to a new location. Types of transformation are reflection, rotation, translation and dilation.
Dilation is the increase or decrease in the size of a figure.
Figure A maps to figure B with a scale factor of 4/9, hence:
4/9 * x = 16
x = 54
Figure A maps to figure B with a scale factor of 4/9, hence the value of x is x = 54
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Kaylen earned $14.87 selling cookies and $7.76 selling
lemonade. then she bought 2 pizzas for $5.00 each.
how much money does she have left?
lemovade
declerc
students, draw anywhere on this slide!
Answer: $12.63
Step-by-step explanation:
$14.87 + $7.76 = $22.63
2pizzas x $5 = $10
$22.63 - $10 = $12.63
NEED HELP ASAPP PLEASEE
Answer:
22.25
Step-by-step explanation:
take the two values away
the sector is the full address , the triangle is in it. the segment is the bit left near the circumference
According to the 2010 Census, 13% of all housing units in the United States were vacant. A Vigo County supervisor wonders if her county is different from this. She randomly selects 550 housing units in her county and finds that 130 of the housing units are vacant. Conduct an appropriate hypothesis test for the supervisor and state your conclusion.
The conclusion is that there is sufficient evidence to support the claim that the percentage of vacant housing units in their district differs from the percentage of vacant housing units in the country.
Given that the total number of a housing units in her country is 550 and the number of vacant housing units in her country is 130.
The Null hypothesis is p₀=13%=0.13, n=550 and x=130
Significance level α=0.05
Claims given: percentage other than 13%
The claim is anyone of the null hypothesis or the alternative hypothesis. The null hypothesis shows that the proportion of the population is equal to the value stated in the claim. If the null hypothesis is a claim, then the alternative hypothesis states the opposite of the null hypothesis.
H₀:p=13%=0.13
Hₐ:p/0.13
I'm interested in testing population proportion claims, so I'll use z-text with proportions.
Conditions: Random sample, 10% condition, pass / fail condition
Sample: I'm happy because the housing units were randomly selected. 10% Condition: We are happy because the sample of 550 homes in your county is less than 10% of the population.
Pass/ fail condition: Satisfied because both np₀ and n(1-p₀) are both at least 10.
np₀=550(0.13)=71.5≥10
n(1-p₀)=550(1-0.13)=478.5≥10
Therefore, we conclude that all the conditions are met.
The sample percentage is the number of successes divided by the sample size.
[tex]\begin{aligned}\hat{p}&=\frac{x}{n}\\ &=\frac{130}{550}\\ &\approx 0.2364\end[/tex]
Determines the value of the test statistic:
[tex]\begin{aligned}z&=\frac{\hat{p}-p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} \\ &=\frac{0.2364-0.13}{\sqrt{\frac{0.13(1-0.13)}{550}}}\\ &=\frac{0.1064}{0.0143}\\&\approx 7.41 \end[/tex]
The P-value is the probability of getting a test statistic value or a more extreme value if the null hypothesis is true. Use the normal probability table in the appendix to determine the P-value.
P=P(Z<−7.41 or Z>7.41)
P=2P(Z>7.41)
P=2(0)
P=0
If the P-value is less than the significance level α, the null hypothesis is rejected.
P<0.05⇒ Reject H₀
Therefore, there is ample evidence to support the claim that the proportion of vacant homes in the area is different from the proportion of vacant homes in the country.
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Help me with answer to these questions and sorry that it is in spanish.
The computation of the decimals are illustrated below.
How to calculate the values?1. 7/4 + √5
= 1.75 + 2.24
= 3.99
2. -✓2 + 9
= -1.41 + 9
= 7.59
3. 2 - 5/4
= 2 - 1.25
= 0.75
4. 7.93 + 1/7
= 7.93 + 0.14
= 8.07
5. π + ✓5
= 3.14 + 2.24
= 5.38
6. ✓2 + 13/4 - 3
= 1.41 + 3.25 - 3
= 1.66
7. 8.5 - 12.65
= -4.15
8. 4 + ✓3 - 11/3
= 4 + 1.73 - 3.67
= 2.06
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Enter the first 4 terms of the sequence defined by the given rule. assume that the domain of each function is the set of whole numbers greater than 0. f(n) = (2n + 2)2 the first 4 terms of the sequence are , , , and .
The first four terms of the sequence are 8,12,16 and 20.
What is a Sequence?A sequence is an enumerated group of items in mathematics where repetitions are permitted and order is important. Similar to a set, it has members (also called elements, or terms). The length of the series is the number of elements (potentially infinite). In contrast to a set, the same items might appear more than once in a sequence at various points, and unlike a set, the order is important. A sequence can be described formally as a function from natural numbers (the positions of the sequence's elements) to the items at each of those positions. An indexed family, which is a function from an index set that may not be a set of numbers to another set of elements, can be thought of as a generalization of the idea of a sequence.Given the function is f(n) = (2n+2)2
Now, we want first four terms, therefore, putting 1, 2, 3, 4 in the sequence we get:
f(1) = (2*1+2)2 = 8
f(2) = (2*2+2)2 = 12
f(3) = (2*3+2)2 = 16
f(4) = (2*4+2)2 = 20
Hence, The first four terms of the sequence are 8,12,16 and 20.
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Please help!!! Having some trouble with this! Tom, Jim, and Matt want to build a castle next to their swing set. The castle is as wide as the swing set, and 6 feet long. The additional play area is enlarged to yield the following layout:|
Length = 7 + 2x [L]
Breadth = 6 + x [B]
So, Area = L × B
=> Area = (7 + 2x) (6 + x)
=> Area = 7 × 6 + 7x + 2x × 6 + 2x²
=> Area = 42 + 7x + 12x + 2x²
=> Area = 2x² + 19x + 42
Option B is correct.
Any questions, you may ask.
Answer: b.) 2x² + 19x + 42
Area of rectangle:
⇒ Length × Width
⇒ (6 + x) × (7 + 2x)
apply distributive method: (a + b)(c + d) = ac + ad + bc + bd⇒ 6(7) + 6(2x) + x(7) + x(2x)
simplify
⇒ 42 + 12x + 7x + 2x²
arrange
⇒ 2x² + 19x + 42
A retail shop accepts only cash or checks. Suppose that 60% of its customers carry cash, 59% carry checks, and 77% carry cash or checks (or both). What is the probability that a randomly chosen customer at the shop is carrying both cash and checks?
[tex]p(c) = 0.6 \\ p(ch) = 0.59 \\ p(any)= 0.77[/tex]
[tex]p(both) = p(c) + p(ch) - p(any) \\ p(both) = 0.6 + 0.59 - 0.77 \\ p(both) = 0.42[/tex]
mark the congruent parts of the triangles and complete the proof
1) given
2) definition of an angle bisector
3) reflexive
4) [tex]\triangle ABD \cong \triangle CBD[/tex] (ASA)
5) [tex]\angle A \cong \angle C[/tex] (CPCTC)
which factor tree for the graph the function below? Check all that apply.
f(x)= 4•5^x
The statements that are true are B, C, and F.
Which facts are true about the exponential function?
Here we have the exponential function:
[tex]f(x) = 4*5^x[/tex]
We can see that the initial value is 4, and the base is 5. Because the base is larger than 1, this is an exponential growth function.
The statements that are true are:
B: The domain of an exponential function is always the set of all real numbers.
C: Is an exponential growth, so yes, it is increasing.
F: The y-intercept is given by evaluating the function in x = 0, when you do that, you get the initial value, which is 4. So the y-intercept is (0, 4).
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Area=
Help me please!! Thanks :)
Answer:
80π
Step-by-step explanation:
Circle O with radius IO:
large radius = IO = 12 = R
Circle O with diameter JK
IJ = JK = KL = 2 × IO = 24
IJ = JK = KL = 24/3 = 8
small radius = 8/2 = 4 = r
The shaded area is a semicircle of radius 12 minus a semicircle of radius 4 plus two semicircles of radius 4.
That is the same as
1 semicircle of radius 12 plus 1 semicircle of radius 4
A = πR²/2 + πr²/2
A = π(12² + 4²)/2
A = 160π/2
A = 80π
The data show the heights in feet of 14 roller coasters. find the mean, median, midrange, and mode for the data.
The mean of the heights of roller coaster is 27.92 feet, median is 21, mode is 16 and mid range is 35.5.
Given heights in feet of 14 roller coaster 11,12,13,14,16,21,25,26,31,40,51,55,60.
We have to calculate the mean, median, mid range and mode for the data.
Mean is the sum of numbers divided by the numbers.
Mean=∑X/n
∑X=11+12+13+14+16+21+25+26+31+40+51+55+60
=391
n=14
Mean=391/14
=27.92 feet.
Median is the central value of the data given.
Median=(n/2)th term ,when n is even, ( n/2)th term, (n+1)/2th term when n is odd.
Median=14/2th term=7th term which is 21.
Mode is the number which has higher frequency.
Mode is 16 because it comes 2 times.
Mid range is the sum of upper value and lower value of data divided by 2.
Mid range=(11+60)/2==35.5.
Hence the mean is 27.92 feet, median is 21 feet, mid range is 35.5 feet, mode is 16 feet.
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Question is incomplete as the values of height should be included as under:
11,12,13,14,16,16,21,25,26,31,40,51,55,60.
The value represents the number of times the interest is compounded in a year.
The compound periods and Interest rates for compound interest have been explained below.
How to define compound interest terms?
1) Compound Periods (n); If we want to calculate the compound interest, the number of compounding periods makes a significant difference due to the fact that the higher the number of compounding periods, the greater the amount of compound interest.
2) Interest rate per compounding periods (I); This affects the interest rate compounded annually due to the fact that the interest rate will be compounded once a year.
Complete question is;
In your own words and using visuals, explain the following. Regarding compound interest, how does the number of times interest is compounded per year affect the:
a) number of compounding periods (n)
b) interest rate per compounding period (i)
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ista father has an age that is 2 years greater than four times her age.The sum of their ages is 57 yeasrs
Answer:
Ista is 11 years old. Her father is 46 years old.
Step-by-step explanation:
Let ista's age be x.
Fathers age=4x+2.
Fathers age + Ista's age = 57.
4x+2+x=57.
Combine like terms:
5x+2 -2 =57 -2
5x=55
x=11
Fathers age: 4(11)+2
= 46
If ista father has an age that is 2 years greater than four times her age. The sum of their ages is 57 years. Then ista age is 11 and her father age is 46.
What is Equation?Two or more expressions with an equal sign is called as equation.
Given that,
ista father has an age that is 2 years greater than four times her age.The sum of their ages is 57 yeasrs
Let the age of ista be x.
ista father has an age that is 2 years greater than four times her age.
it means father age = 2+4x
The sum of ista age and her father age is 57.
So x+(2+4x)=57
Add the common terms
5x+2=57
Subtract 2 from both sides
5x=55
Divide both sides by 5
x=11
Hence ista age is 11 and her father age is 46.
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A marketing firm wants to estimate how much root beer the average teenager drinks per year. A previous study found a standard deviation of 1.24 liters. How many teenagers must the firm interview in order to have a margin of error of at most 0.4 liter when constructing a 99% confidence interval
At least 832 teenagers must be interviewed.
It is given that the standard deviation is of 1.24 liters.
We have that to find our level, that is the subtraction of 1 by the confidence interval divided by 2. So:
[tex]\alpha = \frac{1-0.99}{2}[/tex] = 0.005
Now, we have to find z in the Ztable as such z has a p-value of [tex]1-\alpha[/tex].
So it is z with a p-value of [tex]1-0.005 = 0.995[/tex], so [tex]z= 2.575[/tex]
Now, find the margin of error M as such
[tex]M = Z * \frac{SD}{\sqrt{n} }[/tex]
In which SD is the standard deviation of the population and n is the size of the sample.
How many teenagers must the firm interview in order to have a margin of error of at most 0.1 liters when constructing a 99% confidence interval?
At least n teenager must be interviewed.
n is found when M = 0.1.
We have that SD = 1.12
So
[tex]0.1 = 2.575*\frac{1.12}{\sqrt{N} }[/tex]
[tex]\sqrt{n} = \frac{2.575*1.12}{0.1}[/tex]
[tex]n = 831.7[/tex]
So rounding up to at least 832 teenagers must be interviewed.
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