The given infinite geometric series 2/3+4/9+8/27+ ...... is convergent and the sum is 2
The given geometric series is:
2/3 + 4/9 + 8/27 + ......
The common ratio (r) of a geometric series is given by:
r = a(n) / a(n - 1)
Notice that in the given series:
a(1) = 2/3
a(2) = 4/9
a(3) = 8/27
Hence, the common ratio is
r = a(2) / a(1) = (4/9) / (2/3) = 2/3
An infinite geometric series is convergent if 0 < | r | < 1.
Since r = 2/3, then the series is convergent and the sum exists.
Use the sum formula:
S = a(1) / (1 - r)
Substitute a(1) = 2/3 and r = 2/3 into the formula:
S = (2/3) / (1 - 2/3)
= (2/3) / (1/3)
= 2
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Plot two numbers that are 5 units from zero on the number line.
The two numbers which are 5 units away from zero on the number line as required in the task content are; -5 and 5.
Which two numbers are 5 units away from 0 on the number line?It follows from the task content that the numbers which are 5 units away from zero on the number line be identified and plotted.
Hence, given that the numbers in discuss are represented by; x; we have;
| x - 0 | = 5
| x | = 5
Ultimately, x = ±5. where +5 is to the right and -5 is to the left of zero respectively.
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Find the diameter and radius of a circle with the given circumference. Round to the nearest hundredth.
C = 124ft
A circle is a curve sketched out by a point moving in a plane. The radius and the diameter of the circle are 19.73 ft and 39.46 ft, respectively.
What is a circle?A circle is a curve sketched out by a point moving in a plane so that its distance from a given point is constant; alternatively, it is the shape formed by all points in a plane that are at a set distance from a given point, the center.
Given that circumference of the circle as 124 ft. Therefore, the radius and the diameter of the circle can be written as,
Circumference of circle = 2 × π × (Radius of the circle)
124 ft = 2 × π × (Radius of the circle)
(Radius of the circle) = 124 ft/ (2 × π)
The radius of the circle = 19.73 ft
Diameter of circle = 2 × (Radius of the circle)
= 2 × 19.73 ft
= 39.46 ft
Hence, the radius and the diameter of the circle are 19.73 ft and 39.46 ft, respectively.
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#1: Jane has 6 feet of ribbon. If one
meter is about 3 feet, how many
meters of ribbon does Jane have?
O 12
02
O
O 1
Answer: jane has 1.5 meters of ribbon
Step-by-step explanation:
Gio sells gelato and collected sales data for the past few days. gio wants to use the naive method to forecast. calculate the naive forecast for day 6. day 1 2 3 4 5 sales 90 97 92 95 93
The naive forecast for day 6 is 94.
What is a naive forecast?A method of estimation that uses the last period's actuals as the forecast for a current period without making any adjustments or attempting to identify the causes. It is solely used to compare forecasts using more advanced (better) approaches.The calculation of an angle histogram using the naive assumption that the accumulation of points corresponding to the directions of interest will produce peaks that may be seen.There are three fundamental categories: causal models, time series analysis and projection, and qualitative approaches.The main four quantitative budget forecasting techniques—straight line, moving average, simple linear regression, and multiple linear regression—are covered in this article despite the fact that there are many other regularly used quantitative budget forecasting tools.
The naive forecast for the nth period is calculated as follows.
Forecast period Yn=Yn−1
Period Sales Naive Forecasts
1 90 -
2 97 90
3 92 97
4 90 92
5 94 90
6 94
Hence, the naive forecast for day 6 is 94.
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Silas pays 8% interest on his $22,000 college loan and 11% interest on his $10,000 car loan. What average interest rate does he pay on the total $32,000 he owes? Round your answer to the nearest tenth of a percent.
The average interest rate does he pay on the total $32,000 he owes is 9. 5%
How to determine the average
From the information given, we have that;
8% interest on his $22,00011% interest on his $10,000Given the total pay he owes as;
= $22, 000 +$ 10, 000
= $32, 000
The average interest rate is expressed as;
= sum of interest rate for college loan and car/ number of interest rate
Substitute the values
= 8 + 11/ 2
= 19/ 2
Find the quotient
= 9. 5%
Thus, the average interest rate does he pay on the total $32,000 he owes is 9. 5%
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a. Use your calculator to generate an arithmetic sequence with a common difference of -7 . How could you use a calculator to find the 6th term? The 8th term? The 20th term?
The 20th term of the arithmetic sequence is 122.
What is arithmetic sequence?nth term = a1 + d(n - 1) so here:
20th term = -30 + 8(20 - 1)
- 30 + 152
[tex]a_{20}[/tex] = 122.
An ordered group of numbers with a shared difference between each succeeding word is known as an arithmetic sequence. For instance, the common difference in the arithmetic series 3, 9, 15, 21, and 27 is 6.
Arithmetic sequences are sequences containing these patterns. The distance between succeeding terms in an arithmetic series is always the same. The difference between consecutive words is always two, hence the sequence 3, 5, 7, 9... is arithmetic.
An explicit formula that states a = d (n - 1) + c, where d is the common difference between succeeding words, and c = a1, can be used to establish an arithmetic sequence.
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The arithmetic mean of two terms in an arithmetic sequence is 42 . One term is 30 . Find the other term.
The arithmetic mean of two terms in an arithmetic sequence is 42 .
What is sequence?One term is 30 then other term is
42 = (x+30)/2
x=54
An ordered group of numbers with a shared difference between each succeeding word is known as an arithmetic sequence. For instance, the common difference in the arithmetic series 3, 9, 15, 21, and 27 is 6.
Arithmetic sequences are sequences containing these patterns. The distance between succeeding terms in an arithmetic series is always the same. The difference between consecutive words is always two, hence the sequence 3, 5, 7, 9... is arithmetic.
An explicit formula that states a = d (n - 1) + c, where d is the common difference between succeeding words, and c = a1, can be used to establish an arithmetic sequence.
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. If AD=26, BF=8, and DF=8, then what is the value of AB?
By using collinearity assumptions and addition and subtraction of line segments, the length of the line segment AB is 26 units.
What is the length of missing line segment?
Let A, B, D, F be points lying on the same line such that A(x) ≤ B(x) ≤ D(x) ≤ F(x), where x is the position of each point on a number line. Then, the value of the length of the line segment AB is:
AB = AD + DF - BF
AB = 26 + 8 - 8
AB = 26
Please notice that B(x) = D(x) as AB = AD.
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I need help please I’m lost
Angle Pairs and their Relationships HW
4)
The measures of the angles m∠1 and m∠2 are: m∠1 = 109 and m∠2 = 71
What is Linear pair angles ?Linear pair angles are two angles which are adjacents and supplementary.
Given that: m∠2 and m∠1 form a linear pair
we know that the angles and are supplementary, which means that they add up to 180 degrees.
In order to solve for "x," we can write the following expression:
180 = (5x + 9) + (3x + 11)
8x = 180 - 20
x = [tex]\frac{160}{8}[/tex]
x = 20
Therefore, substituting, we get that the measures of the angles and are
m∠1 = 5(20) +9 = 109
m∠2 = 3(20) + 11 = 71
5)
the angle m∠2 = 86°
Given that m∠1 and m∠2 are vertical angles and m∠1 = (17x + 1) and m∠2 = (20x - 14)
Vertical angles are always congruent
m∠1 = 17x + 1
m∠2 = 20x - 14
Since ∠1 and ∠2 are vertical:
m∠1 = m∠2
17x + 1 = 20x - 14
Collect like terms
20x - 17x = 1 + 14
3x = 15
Divide both sides by 3
3x / 3 = 15/3
x = 5
m∠2 = 20x - 14
m∠2 = 20(5) - 14
m∠2 = 100 - 14
m∠2 = 86°
6)
The value of each angle that is p=119 and q=61
In light of this, the measure of an angle is P, which is three less than the measure of angle q.
P + q = 180 ( By definition of supplementary)
p = 2q -3
Substitute the value
Then ,we get
2q - 3 + q =180
3q = 180 + 3
3q = 183
q= [tex]\frac{183}{3}[/tex]
= 61
then, if you substitute q's value, you get
p= 2(61) - 3 = 122 - 3 = 11.9
Hence, p= 119 and q= 61
7)
The value of x is 19°
Given that : ∠DBE =2x - 1
∠CBE =5x – 42 and BD is perpendicular to AC.
A right angle is formed when two lines are perpendicular.
∠DBE + ∠CBE = 90
90 = (2x - 1) + (5x - 42)
7x = 133
x = 19°
8)
The value of x and y are : x= 26° and y= 9°.
As a linear pair, (5x - 17)° + (3x - 11)° = 180°.
or, 8x - 28°= 180°
8x = 180° + 28°
Therefore, the value of x is 26°.
Now,
3x - 11°= 3×26°-11° = 67°
Again,
67°+90°+(2y+5)° = 180° { being linear pair}.
157°+2y+5° = 180°
or, 2y + 162° = 180°
or, 2y = 180° - 162°
y = [tex]\frac{18}{2}[/tex]
Therefore, the value of y is 9°.
The value of x is 26° and y is 9°.
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Write the equation of each line in slope-intercept form. y/9 + x/3 = 2
The Slope-intercept form of the given equation is y = -3x + 18 .
An equation in Slope-intercept form is of the form
[tex]y = mx + b[/tex]
where m is the slope and b is the y intercept.
Now, the given equation is (y/9) + (x/3) = 2
Taking LCM on the left hand side of the equation, we get
⇒ [tex](3y+9x)/(2y) = 2[/tex]
⇒ 3y + 9x = 54
⇒ 3y = 54 - 9x
Dividing both side of the equation by 3, we get
⇒ y = [tex](54 - 9x)/3[/tex]
⇒ y = 18 - 3x
⇒ y = -3x + 18
∴ The equation y = -3x + 18 is in Slope-intercept form.
In the equation the slope is -3 and the y-intercept is 18.
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In the Mars Cars video game, players gain and lose points by avoiding or hitting obstacles on a martian race track. First, Gary gained 20 points by jumping over a crater. Then, he lost 45 points for getting stuck in a dust storm.
What was Gary's score after getting stuck in the dust storm?
The Gary's score after getting stuck in the dust storm will be -35 points.
It is given in the problem that,
The points gained by Gary by jumping over a crater = 20 points
And then,
The points lost by Gary for getting stuck in a dust storm = 45 points
We need to find,
The Gary's score after getting stuck in the dust storm
As in the beginning of the game, Gary's account had 20 points but as he then stuck in the dust storm he lost 45 points, which means he has lost points more than he had. So, now Gary would have negative points left in his account in the game
= 10 - 45
= - 35
Hence, the Gary's score after getting stuck in the dust storm will be -35 points.
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i need help fast pleeeeas hurry
6. What's the product of 3 2/3 and 14 2/5 ?
A. 52 415
B. 42 45
C. 54
D. 52 45
Answer:
D. 52 45
Step-by-step explanation:
x2+6x−16 vertex form
Answer:
Hello,
Step-by-step explanation:
[tex]x^2+6x-16\\\\=x^2+2*3x+3^2-9-16\\\\=(x+3)^2-25\\[/tex]
8/3= 19/6n - 1/2n
I NEED HELP
Answer:
n = 1
Step-by-step explanation:
[tex]\frac{8}{3}[/tex] = [tex]\frac{19}{6}[/tex] n - [tex]\frac{1}{2}[/tex] n
multiply through by 6 ( the LCM of 3, 6 and 2 ) to clear the fractions
16 = 19n - 3n
16 = 16n ( divide both sides by 16 )
1 = n
Do the equations ;x + 2 = 6
and ;(*+ 2) = 6 represent the same situation?
Explain.
The given equations have in one, (1/5) is the coefficient of x, in the second equation, (1/5) is the coefficient of (x + 2), while both equations equals 6. Therefore, the value of x is different in both equations which do not represent the same situation.
How the given equations be evaluated?The given equations are presented as follows;
[tex]\frac{1}{5} \cdot x + 2 = 6...(1)[/tex]
[tex] \frac{1}{5} \cdot (x + 2) = 6...(2)[/tex]
Solving each equation for x gives;
For equation (1), x = 5×(6 - 2) = 20
x = 20
For equation (2), we have;
x = 6 × 5 - 2 = 28
x = 28
Given that the value of x in the given equations are different, the situations are different.
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Under her cell phone plan, Caroline pays a flat cost of $54.50 per month and $5 per gigabyte. She wants to keep her bill at $74 per month. Write and solve an equation which can be used to determine gg, the number of gigabytes of data Caroline can use while staying within her budget.
Answer:
5 gg
Step-by-step explanation:
Let us treat the number of gigabytes of data she can use (at max) per month as x.
We can say that [tex]54.5 + 5x = 74[/tex] will always be deducted from her and $5x per month for the extra data she uses.
[tex]54.5 + 5x = 74[/tex]
⇒ [tex]5x = 74 - 54.5[/tex]
⇒ [tex]5x = 25.5[/tex]
⇒ [tex]x = 25.5 / 5[/tex]
⇒ [tex]x = 5.1[/tex]
As she can only buy a fixed number of data, which cannot be a decimal, let us take the number directly below 5.1 - which is 5 - as the answer.
Caroline can use up to 5gg of data
Write a definite integral that represents the area of the region. (do not evaluate the integral.) y1 = x2 2x 3 y2 = 2x 12
The definite integral that represents the area of the region under the given curves is: [tex]\int\limits^3_{-3} {x^2-9} \, dx[/tex]
What is the area of the region under a curve?By performing a definite integral between the two locations, one can determine the area under a curve between two points. Integrate y = f(x) between the limits of a and b to determine the area under the curve y = f(x) between x = a & x = b. With the specified limits, integration can be used to calculate this area.Given:
[tex]y_1=x^2+2x+3[/tex][tex]y_2=2x+12[/tex]The points where the two intersect will be given by:
y₁ = y₂
=> x² + 2x + 3 = 2x + 12
=> x² + 3 = 12
=> x² = 9
=> x = ± 3
For x₁ = 3, y₁ = 2(3) + 12 = 18 => (x₁, y₁) = (3, 18)
For x₂ = -3, y₂ = 2(-3) + 12 = 6 => (x₂, y₂) = (-3, 6)
Now, for the area of the region under first curve [tex]y_1=x^2+2x+3[/tex]:
A₁ = [tex]\int\limits^3_{-3} {x^2+2x+12} \, dx[/tex]
For the area of the region under the second curve [tex]y_2=2x+12[/tex]:
A₂ = [tex]\int\limits^3_{-3} {2x+12} \, dx[/tex]
For the required area of the region bounded by the two curves will be given by:
A = A₂ - A₁ = [tex]\int\limits^3_{-3} {x^2+2x+3 - (2x +12)} \, dx[/tex]
A = [tex]\int\limits^3_{-3} {x^2-9} \, dx[/tex]
Refer to the image for the area so bounded by the curves.
Hence, The definite integral that represents the area of the region under the given curves is: [tex]\int\limits^3_{-3} {x^2-9} \, dx[/tex].
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Mt.Whitney is the highest peak in California at 14,949 feet. The city of El Centro, also in California, is 39 feet below sea level. What is the difference in elevation between in Mt. Whitney’s peak and city of El Centro?
Answer:
if the peak level is 14949 above sea level , so the difference in elevation is 14949 + 39 =14988 feet
Write 18x²-2/ 3x²-5x - 2} in simplest terms.
F. 18/ 3x+1
G. 2(3 x+1)/x-2
H. 2(3 x-1)/x-2
J. 2(3 x-1)
18/ 3x+1 is 18x²-2/ 3x²-5x - 2} in simplest terms.
What are examples of linear equations?
Ax+By=C is the usual form for two-variable linear equations. A standard form linear equation is, for instance, 2x+3y=5. When an equation is given in this format, finding both intercepts is rather simple (x and y). When attempting to solve systems involving two linear equations, this form is also quite helpful.18 x² - 2 / 3x² - 5x - 2
3x² - 5x - 2 = 3x² - ( 6- 1 ) x - 2 = 0
⇒ 3x² - 6x + x - 2 = 0
⇒ 3x( x - 2 ) + 1 ( x - 2) = 0
⇒ ( 3x + 1 ) ( x - 2 ) = 0
put in equation -
= 18 x² - 2 / ( 3x + 1 ) ( x - 2 )
= 18/ 3x+1
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What is the average rate of change, please simplify your answer (PICTURE BELOW)
Answer:
[tex]{ \tt{change = \frac{y_{b} - y _{a} }{x _{b} - x _{a} } }} \\ \\[/tex]
xb is 4; yb is 16 [from graph (4, 16)]xa is 1; ya is 2 [from graph (1, 2)][tex]{ \tt{ change = \frac{16 - 2}{4 - 1} }} \\ \\ = { \tt{ \frac{14}{3} }} \\ \\ = 4 \frac{2}{3} [/tex]
Ava's sister is 9 years less than twice Ava's age b
Answer:
2b + 9
Hope this helps :)
Step-by-step explanation:
This equation shows the statement, "Ava's sister is 9 years less than twice Ava's age b."
Consider the functions f(x)=4^x and g(x)=x-2. The function y is defined as y=f(g(x)) state the equation for y.
the equation for y will be y = 4ˣ / 16 when the function y is defined as y = f ( g ( x ) ).
We are given the functions:
f ( x ) = 4ˣ
g ( x ) = x - 2
Now, we have a function y defined as:
y = f ( g ( x ) )
Now, we need to find the equation for y.
We know that:
f ( g ( x ) ) = 4⁽ ˣ⁻ ² ⁾
f ( g ( x ) ) = 4ˣ / 4²
f ( g ( x ) ) = 4ˣ / 16
Therefore, we get that, the equation for y will be y = 4ˣ / 16 when the function y is defined as y = f ( g ( x ) ).
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Read the problem. Identify what you need to know. Then organize the data to solve the problem.
There are 10 sophomore, 8 junior, and 9 senior members in student council. Each member is assigned to help plan one school activity during the year. There are 4 sophomores working on the field day and 6 working on the pep rally. Of the juniors, 2 are working on the field day and 5 are working on the school dance. There are 2 seniors working on the pep rally. If each activity has a total of 9 students helping to plan it, what is the probability that a randomly selected student council member is a junior or is working on the field day?
F. 1/5
G. 4/18
H. 5/9
J. 2/3
Using the combination formula, the committee can be selected in 1,211,760 ways.
What is the probability?Probability refers to a possibility that deals with the occurrence of random events.
The probability of all the events occurring need to be 1.
Combination formula is the number of different combinations of x objects from a set of n elements.
In this problem:
There are 10 sophomore, 8 junior, and 9 senior members in student council.
They are independent, so we can just multiply them, thus;
T = 18!/ 2! 16! x 12!/ 2! 10! x 10!/ 3! 7!
T = 1,211,760
Hence, The committee can be selected in 1,211,760 ways.
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For each system, choose the method of solving that seems easier to use. Explain why you made each choice. Solve each system.
6x - 3y = 3 , 5x - 5y = 10
By elimination, the solution of the system of equations, 6x - 3y = 3 and 5x - 5y = 10, is (-1 , -3).
A system of equations is a set of two or more equations which includes common variables. To solve system of equations, we must find the value of the unknown variables used in the equations that must satisfy all the equations.
There are three methods that can be used to solve system of equations.
1. Elimination
2. Substitution
3. Graphing
First, simplify both equations by dividing the first equation by 3 and the second equation by 5.
6x - 3y = 3 ⇒ 2x - y = 1 (equation 1)
5x - 5y = 10 ⇒ x - y = 2 (equation 2)
Use the elimination method since subtracting the two equations will eliminate the variable y.
2x - y = 1 (equation 1)
x - y = 2 (equation 2)
x = -1
x = -1
Substitute the value of x to any of the two equation and solve for y.
6x - 3y = 3 (equation 1)
6(-1) - 3y = 3
-3y = 3 + 6
-3y = 9
y = -3
Hence, the solution of the system of equations is (-1 , -3).
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What is the solution to the linear equation?
-12+36-1--5-b
O b=-2
O b=-1.5
O b= 1.5
O b=2
Answer:
where is the equal to sign????
c. Find the 9 th term of the geometric sequence from part (b).
The 9th term of the geometric sequence can be determined by using the nth term of geometric sequence formula: an=arn-1
Geometric sequence is the series of number in which each term is determined by multiplying the common ratio with the preceding term.
In the formula, an=arn-1, a is the first term, n is the number of terms, and r is the common ratio.
The geometric sequence from the part b is 2,4,8,…
The first term a = 2
The common ratio r = 2
The number of term n = 9
Hence, to find the 9th term,
a9=2(2)9-1
a9=2(2)8
a9=512
The 9th term of the geometric sequence 2,4,8,… is 512
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Name the geometric terms modeled by a wall and the floor. (Blank 1) Blank 1 options two lines intersecting in a point • two lines intersecting in a line • two planes intersecting in a line three planes intersecting in a point
Answer: Two planes intersecting in a line
Explanation:
The wall and floor are two different planes. They meet up to form a line which is the crease between them.
A plane is a flat surface without any bends or curves to it. In theoretical terms, planes go on forever in all directions. Realistically there are limits of course. The wall or floor cannot go on forever.
Another example of two planes would be found in a book. Each page is a plane. Two adjacent pages meet up to form a line (i.e. the spine of the book is a line).
Jim thinks that the value of a baseball card can be modeled by a decay formula and that the value will decrease at a rate of 0.2% each year. The card was originally valued at $250 in 2007.
What would Jim expect the value of the card to be in 2018?
Answer: $244.55
A = $250 ; r=0.002 t= 11 [From 2007 to 2018 , t=2018-2007]
PLEASE I NEED HELP QUICK EASY POINTS
Answer: f(-1) = -12
Step-by-step explanation:
To solve, we will substitute t for -1 into the function.
Given:
h(t) = (t + 7)(t - 1)
Substitute:
h(-1) = (-1 + 7)(-1 - 1)
Add and subtract:
h(-1) = (6)(-2)
Multiply:
f(-1) = -12
