The length of each side of the octagon with area of 392 square centimeters and an apothem of 14 centimeters is 7 centimeters .
Area of an Octagon:Compared to lower sided shapes, an octagon's area is a little more complicated. A normal octagon's area is equal to 4 as. Where "s" stands for the octagonal side and "a" for the apothem, which runs straight from the pentagon's center to one of its sides.
The formula for the area of a regular octagon is:
A = 4 × apothem × side
Using this formula and the given values, we can solve for the length of each side of the octagon:
392 = 4 × 14 × side
Dividing both sides by 4 × 14, we get:
side = 392 ÷ (4 × 14)
Simplifying the right-hand side, we get:
side = 392 ÷ 56 = 7 cm.
Therefore, the length of each side of the octagon is 7 cm .
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The three boys in Josh's family own a total of 28 games. The youngest, Kyle has 8 games. Josh has 4 times as many games as his older brother, Marcus. How many games does Josh own?
A 20 games
B 16 games
C 12 games
D 4 games
1. Find an equation of the line that satisfies the given conditions.
Through (−5, 2); perpendicular to the line y = − 1/2x + 4
2. Find an equation of the line that satisfies the given conditions.
y-intercept 3; parallel to the line 3x + 4y + 2 = 0
3. Find an equation of the line that satisfies the given conditions.
Through (−5, −13); perpendicular to the line passing through (−2, −1) and (2, −3)
Through (−5, 2); perpendicular to the line y = − 1/2x + 4 will be ⇒ y = 2x + 12 ⇒ y= 2x -3
1. y= - 1/2 x + 4
passes through (-5.2)
line perpendicular to given line
⇒ (slope of given line) * (slope of newline) = -1
(- 1/2) * (m) = -1
∴ m =2
equation of line passing through (-5 , 2) having slope m = 2 is
y- [tex]y_{1}[/tex] = slope ([tex]x-x_{1}[/tex])
y-2 = 2(x-(-5))
y-2 = 2(x=5)
y-2 = 2x + 10
∴ y = 2x + 12
2. parallel to line : 3x + 4y +2=0
⇒ slope = -3/4
4y = -3x -2
∴ y= -3/4x -1/2
y-intercept = 3
since equation of line having slope & y-intercept is, y =mx+c
slope = -3/4 ; y-intercept = 3
y=mx+c
y=-3/4x+3
4y = -3x + 12
∴ 3x =4y = 12
3. line passing through : (-5, -13) & perpendicular to line passes through (-2, -1) & (2, -3)
∴ slope of perpendicular line = [tex]y_{2} - y_{1} / x_{2} -x_{1}[/tex]
= -3 - (-1)/ 2 - (-2) = -3 + 1/4 = -2/4 = -1/2
y- (-13)= 2(x -(-5))
y+13 = 2x + 10
⇒ y= 2x -3
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(1 point) Let A = | | 1-1-1-1 | . Find a non-zero vector x in the null space of A.
The null space of A is spanned by the vector x = (t, -t, -t, -t), where t is a real number
Vectors are an important mathematical tool for representing data, equations, and other mathematical objects. In this problem, we are given a matrix A and asked to find a non-zero vector x in the null space of A.
The matrix A = | | 1-1-1-1 | can be reduced to row-echelon form using row operations. The resulting matrix is A = | | 1 0 0 0 | , which has a single non-zero row, indicating that the null space of A is one-dimensional.
Therefore, the non-zero vector x in the null space of A can be written as x = (t, -t, -t, -t), where t is a real number.
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How do you find the slope of the tangent line to the parabola y=7x−x2at the point (1,6)?
what type of effect can outliers have on a regression line?
A regression line is a line of means, and outliers have a big effect on the regression line.
Regression:
Regression is a statistical method used in finance, investing, and other disciplines that attempts to determine the strength and character of the relationship between a dependent variable (usually denoted by Y) and a set of other variables (called variables independent). Also known as simple regression or ordinary least squares (OLS), linear regression is the most common form of this technique. Linear regression establishes a linear relationship between two variables based on a line of best fit. So the linear regression is plotted with a straight line and the slope defines how changes in one variable affect changes in the other variable.
Regression Line:
Regression lines are very useful for the forecasting process. The purpose of this line is to describe the relationship between the dependent variable (variable Y) and one or more independent variables (variable X). Using the equation obtained from the regression line, the analyst can predict the future behavior of the dependent variable by entering different values for the independent variable. Regression lines are widely used in the financial industry and in business in general.
The regression line formula is like the following:
Y = a + bX + u
The multiple regression formula looks like this:
Y = a + b₁X₁ + b₂X₂ + b₃X₃ + … + btXt +u.
Y is the dependent variable
X is the independent ones
a is the interception point
b is the slope
u is the residual regression.
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how to convert 152 cm to inches?
On converting the length , 152 centimeter is equivalent to 59.8424 inches .
There are various units of measuring the length , which include , meter (m) , centimeter (cm) , kilometer (km) , inches (in.) ;
In order to convert 152 centimeters (cm) to inches,
We use the conversion factor of 0.3937 inches per centimeter.
So , we multiply the number of centimeters by this conversion factor to get the equivalent length in inches ;
By using this method, we get ;
⇒ 152 cm × 0.3937 inches/cm = 59.8424 inches ;
Therefore, 152 centimeters is equal to approximately 59.8424 inches.
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Line l passes through the points (-8, 8) and (-4, 0). Write the equation of the image of
after a dilation with a scale factor of 1, centered at the origin.
Write your answer in slope-intercept form.
y =
True or false 5. An equilateral triangle has 3 equal sides and 3 acute angles.
trueeeeeeeeeeeeeeeee
What is the equation of the normal line to the curve
y=sin(sinx)
at x=π ? A. y=x−π
B. y=−x−π
C. y=Ï€x
D. y=πx−1
E.y=− π1 x−1
The equation of the normal line to the curve is y = -x + π
To find the equation of the normal line to the curve y = sin(sinx) at x = π, we need to find the slope of the tangent line at x = π and then use that to determine the slope of the normal line. We can find the slope of the tangent line by taking the derivative of y with respect to x and evaluating it at x = π.
y = sin(sinx)
Taking the derivative with respect to x, we get:
dy/dx = cos(sinx) * cosx
At x = π, we have:
dy/dx = cos(sin(π)) * cos(π) = -1 * (-1) = 1
So the slope of the tangent line at x = π is 1. Since the normal line is perpendicular to the tangent line, its slope is the negative reciprocal of 1, which is -1.
Now we need to find the y-intercept of the normal line, which is the point where the line intersects the y-axis. To do this, we can plug in the values of x = π and y = sin(sinx) = sin(sin(π)) = 0 into the equation of the line:
y = mx + b
0 = -1 * π + b
b = π
So the equation of the normal line is:
y = -x + π
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Divide 15a6b4 by 5a2b.
Answer:
Step-by-step explanation:
15a6b4/5a2b
you can scrap the a and b
15*6*4/ 5*2=
now you keep simplifying the fraction
3*6*4/2=
3*6*2=
36
Admission to Overjoy Amusement Park is $20 $ 20 per visit. However, with a season pass, which costs $40 $ 40 , each admission is $12 $ 12 . Complete the table and write a system of equations to represent the admission costs, where a is the number of admissions and c is the total cost. CLEAR CHECK Complete the table. Cost of Admission($ $ ) Number of Admissions Amount Spent on Season Pass($ $ ) Total Cost With season pass a c Without season pass a c Write a system of equations to represent this problem. With a season pass: Without a season pass:
Total cost with pass = $20a
Total cost without pass = $12a + 40
According to the information
We can write a system of equations for this problem,
Equation 1:
Cost with season pass = 12a + 40
Equation 2:
Cost without season pass = 20a
Therefore,
The system of equations that represents this problem is:
12a + 40 = c ...(i)
20a = c ...(ii)
Now making table for it:
Cost Number Amount Spent Total Cost With Total Cost Without
Ad. Ad. Pass ($) Season Pass Season Pass
20 a 0 20a + 0 20a
12 a 40 12a + 40 12a + 40
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3. A bee can fly at an average rate of 7 miles/hour. If a bee flew nonstop
for 18 hours, how far will it have traveled?
If a bee can fly at an average rate of 7 miles per hour and it flew nonstop for 18 hours, we can find the distance it traveled by multiplying its average speed by the time it flew:
distance = speed x time
distance = 7 miles/hour x 18 hours
distance = 126 miles
Therefore, the bee would have traveled 126 miles in 18 hours.
Answer:
126 m (Miles)
Step-by-step explanation:
distance travelled = speed×time = 7×18 = 126 miles
what is standard deviation meaning?
A measure of variation that includes spread, variance, and spread from the mean is the standard deviation.
The skewness of the data is expressed as standard deviation. It measures how far each observation is from the mean. Approximately 95% of the values in each distribution fall within 2 standard deviations of the mean.
Standard deviation is a measure of spread, variance, and deviation from the mean given spread.
The standard deviation represents the "typical" deviation from the mean. This is a popular measure of volatility because it preserves the original units of the data set.
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If triangle XYZ is dilated from center C by a scale factor of 0.5 to form triangle X’Y’Z’, where is the center of the dilation, point c, located?
The center of the dilation, point C, is located at the same point as it was before the dilation. This is because the center of dilation is the point from which the dilation occurs, and it does not change during the dilation process.
To find the coordinates of point C, we can use the properties of dilations. The coordinates of the vertices of the dilated triangle are found by multiplying the coordinates of the original triangle by the scale factor.
So, if the coordinates of point X are (x1, y1), the coordinates of point X' will be (0.5x1, 0.5y1). Similarly, the coordinates of point Y' will be (0.5x2, 0.5y2) and the coordinates of point Z' will be (0.5x3, 0.5y3), where (x2, y2) and (x3, y3) are the coordinates of points Y and Z, respectively.
Since point C is the center of dilation, the coordinates of point C will be the same as the coordinates of the midpoint of the line segment connecting points X and X', points Y and Y', and points Z and Z'. The midpoint of a line segment is found by averaging the coordinates of the endpoints. So, the coordinates of point C will be:
C = ((x1 + 0.5x1)/2, (y1 + 0.5y1)/2) = (0.75x1, 0.75y1)
Therefore, the center of the dilation, point C, is located at (0.75x1, 0.75y1).
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A 25 foot tall flagpole casts a 42 foot shadow. What is the angle the sun hits the flagpole?
The angle the sun hits the flagpole is approximately 33.69°.
Describe angle.An angle is a geometric shape that is defined as the amount of rotation that occurs between two straight lines or planes. Angles are measured in degrees, with 360° representing a full circle.
We need to apply the inverse tangent function to determine the angle at which the sun strikes the flagpole. We are aware that the triangle's adjacent side is 42 feet long and its opposite side is 25 feet tall (the height of the flagpole) (the length of the shadow). This indicates that the following equation should be used to get the angle:
Angle = arctan(25/42)
Therefore, the angle the sun hits the flagpole is approximately 33.69°.
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An insect population after x months can be modeled by the function g(x) = 18(1.3)x. Which statement is the best interpretation of one of the values in this function?
The insect population increased by 13 insects each month.
G.
The insect population decreased by 13 insects each month.
H.
The insect population increased at a rate of 30% each month.
J.
The insect population decreased at a rate of 30% each month.
The side of a cube is multiplied by a factor of 10 . By what factor does the volume of the cube change?
The new volume of the cube is 1000 times the original volume. This means that the volume has increased by a factor of 1000.
What is volume?Volume is a measure of the amount of space that a three-dimensional object occupies. It is a property of a solid shape, and is usually expressed in units of cubic units, such as cubic meters [tex](m^3[/tex]), cubic centimeters [tex](cm^3[/tex]), or cubic feet [tex](ft^3)[/tex].
The volume of a cube is given by the formula [tex]V = s^3[/tex] cube is multiplied by a factor of 10, then the new side length becomes 10s.
The new volume of the cube can be calculated as follows:
V' = [tex](10s)^3[/tex]
= [tex]10^3[/tex] * [tex]s^3[/tex]
= 1000 *[tex]s^3[/tex]
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The cube's new volume is 1000 times larger than its old volume. This indicates that the capacity has multiplied by 1000.
Describe volume?
The quantity of space a three-dimensional object takes up is measured by its volume. It is a characteristic of solid shapes and is frequently stated in cubic unit measurements like cubic meters, cubic centimeters, or cubic feet.
The method for calculating a cube's volume is cube times a factor of 10, where the new side length is 10s.
The new volume of the cube can be calculated as follows:
V' = [tex]s^{3}[/tex]
= [tex](10s)^{3}[/tex]
[tex]V'=1000 s^{3}[/tex]
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(Worth 20 points and will give brainliest if you're right.) Question 2 options: Factor a^3+2a+2a^2+4. Fill in the blanks using the correct sign: (a^2+ ) (a+ )
(please help as soon as you can.)
Answer:
(a^2+2)(a+2)
Step-by-step explanation:
Factor a^3+2a+2a^2+4.
Using Factoring by grouping
a^3+2a + 2a^2+4
Factor out a from the first two terms and 2 from the last two terms.
a( a^2 +2) + 2 ( a^2+2)
Factoring out a^2+2.
(a^2+2)(a+2)
Help! 50 points, Sorry not the best picture…
The completed table in the question used to prove the proportional BD/AB = BC/(AB + AC), when AD is an angle bisector can be presented as follows;
Statement [tex]{}[/tex] Reason
1. EB ║ AD, AD bisects ∠ABC [tex]{}[/tex] 1. Given
2. ∠BEC = ∠DAC [tex]{}[/tex] 2. Corresponding angles theorem
3. ∠C = ∠C [tex]{}[/tex] 3. Reflexive property of congruence
4. ΔBEC = ΔDAC [tex]{}[/tex] 4. AAA similarity postulate
5. DC/AC = BC/EC [tex]{}[/tex] 5. Similar triangles have equivalent ratios
6. EC = EA + AC [tex]{}[/tex] 6. Segment addition postulate
7. DC/AC = BC/(EA + AC)[tex]{}[/tex] [tex]{}[/tex] 7. Similar triangles have equivalent ratios
8. AD/AB = DC/AC [tex]{}[/tex] 8. Triangle angle bisector theorem
9. BD/AB = DC/AC [tex]{}[/tex] 9. Substitution property
10. ∠EBA = ∠DAB [tex]{}[/tex] 10. Alt int. angles theorem
11. ∠DAB = ∠DAC [tex]{}[/tex] 11. Definition of a bisector
12. ∠EBA = ∠BEC [tex]{}[/tex] 12. Alt int. angles theorem
13. ΔEBA is isosceles [tex]{}[/tex] 13. Definition of isosceles triangles
14. CA = AB [tex]{}[/tex] 14. Side length of isosceles
15. BD/AB = BC/EC [tex]{}[/tex] 15. Similar triangles have equal ratios
16. BD/AB = BC/(EA + AC) [tex]{}[/tex] 16. Substitution property
17. EA = AB [tex]{}[/tex] 17. Definition of isosceles triangle ΔEBA
18. BD/AB = BC/(AB + AC) [tex]{}[/tex] 18. Substitution property
What is an angle bisector?An angle bisector is an line that divides an angle into two angles of the same measure.
The details of the some of the reasons useds to prove the specified proportion are presented as follows;
Corresponding angles theorem
The corresponding angles theorem states that the corresponding angles formed between parallel lines and a transversal are congruent
Reflexive property of congruence
The reflexive property of congruence states that a geometric figure, such as an angle is congruent to itself
AA similarity postulate
The AA similarity postulate states that triangles are similar two angles in one triangle are congruent to two angles in the other triangles.
Similar triangles have equivalent ratios
Segment addition postulate
The segment addition postulate states that the point B is located between the points A and C on a segment, only if AC = AB + BC.
Ratios of corresponding sides ar
Triangle angle bisector theorem
The triangle angle bisector theorem states that an angle bisector of a triangle divides the side facing the bisected angle in the ratio of the lengths of the other two sides.
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need help asap
Which ordered pair (from the table in part A) represents the initial value? What does the initial value represent in this problem? (1-2
sentences)
Part D:
What is the rate of change in this equation? What does the rate of change represent in this problem? (1-2 sentences)
Table from part A:
The table of values for y = -128x + 3840 is:
xy
0 3840
5 3200
8 2816
10 2560
30 0
C) The ordered pair that represents the initial value of the function is of (0,3840), meaning that the graph of the function crosses the y-axis at the value of y = 3840.
D) The rate of change of the equation is of -128, meaning that when x increases by one, y is decreased by 128.
How to define a linear function?The following is the definition of a linear function's slope and intercept:
y = mx + b.
In which:
m is the slope, representing the rate of change, that is, by how much y changes when x is increased by one.
The value of y at x=0 is represented by the intercept, b. Hence the ordered pair is of (0,b).
The function is defined as follows for this problem:
y = -128x + 3840.
The slope and intercept are therefore given as follow:
m = -128, b = 3840.
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system developers can initiate a formal project as early as the preliminary investigation stage, or later on, as analysis, design, and implementation activities occur. T/F
True. System developers can initiate a formal project as early as the preliminary investigation stage, or later on, as analysis, design, and implementation activities occur.
When considering the initiation of a formal project, system developers need to assess the risks, costs, benefits, and constraints associated with the project. During the preliminary investigation stage, they conduct a feasibility study to determine whether the project is technically feasible, economically viable, and socially and legally acceptable. If the feasibility study shows that the project is feasible, the developers can initiate the project by defining its scope, objectives, deliverables, and requirements.
Once the project is initiated, the developers can proceed with the analysis, design, and implementation activities. The analysis phase involves gathering and documenting the requirements, analyzing the existing system, and identifying the functional and non-functional requirements of the new system.
True. System developers can initiate a formal project as early as the preliminary investigation stage, or later on, as analysis, design, and implementation activities occur. The decision on when to initiate a formal project depends on several factors, such as the complexity of the system, the available resources, and the level of certainty about the feasibility of the project.
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How much would $10,000 due in 50 years be worth today if the discount rate were 7.5%? a $285.02 b. $266.20 c. S204.36 d. $217.80 e $268.89
The amount would $10,000 due in 50 years be worth today if the discount rate were 7.5% is,
P = $7,042
What is mean by Percentage?
A number or ratio that can be expressed as a fraction of 100 or a relative value indicating hundredth part of any quantity is called percentage.
We know that;
The formula is,
⇒ Amount = P (1 + r)ⁿ
Hence, We get;
10,000 = P (1 + 0.75)⁵⁰
10,000 = P (1.75)⁵⁰
10,000 = P × 1.42
P = 10,000 / 1.42
P = $7,042
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The Tabletop Games club, a student-led campus group, hosted a virtual concert last week to raise awareness about their club and bring community members together. They sold full-price tickets for
$7. 00, tickets for senior citizens were $5. 25
and student tickets cost $5. 50.
After the concert, they counted up the totals. There were a total of 215 tickets sold, and ticket sales brought in $1,214. 00
. It was noted that there were 3 times as many senior citizen tickets sold as there were full-price tickets.
Write and solve a linear system to determine how many of each type of ticket were sold.
There were _____
full-price tickets.
There were ____
senior citizen tickets.
There were ____
student tickets
A system of three linear equations: for the total number of tickets sold is 215 is f + s + t = 215, for the total revenue from ticket sales is $1214 is 7f + 5.25s + 5.5t = 1214, for three times as many senior citizen tickets were sold as full-price tickets is s = 3f.
There were approximately 11 full-price tickets sold. There were 33 senior citizen tickets sold. There were 171 student tickets sold.
Let f, s, t be the number of full-price tickets sold, senior citizen tickets sold, student tickets sold.
Firstly, write a system of three linear equations based on the given information and mark the equation number.
f + s + t = 215 (1) (the total number of tickets sold is 215)
7f + 5.25s + 5.5t = 1214 (2) (the total revenue from ticket sales is $1214)
s = 3f (3) (three times as many senior citizen tickets were sold as full-price tickets)
solve this system by substitution. Using equation (3), we can substitute 3f for s in equations (1) and (2):
f + 3f + t = 215
7f + 5.25(3f) + 5.5t = 1214
Simplifying these equations gives:
4f + t = 215 (4)
24.75f + 5.5t = 1214 (5)
We can solve equation (4) for t to get:
t = 215 - 4f
We can substitute this expression for t in equation (5) and solve for f:
24.75f + 5.5(215 - 4f) = 1214
24.75f + 1182.5 - 22f = 1214
2.75f = 31.5
f = 11.45 ≈ 11
Therefore, approximately 11 full-price tickets were sold. Using equation (3), we can find the number of senior citizen tickets sold:
s = 3f = 3(11) = 33
Therefore, 33 senior citizen tickets were sold. Finally, we can use equation (1) to find the number of student tickets sold:
f + s + t = 215
11 + 33 + t = 215
t = 171
Therefore, 171 student tickets were sold.
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Last month, Ed spent $50 in all. He spent 40% of the money at the movies. How much money did Ed spend at the movies
Answer: I believe the answer would be 20 dollars spent at the movies.
Step-by-step explanation:
please help ASAP!!
(the question above is the previous question)
need help answering #6!!
For each point, their position with respect to its location on the circle are as follows;
(4, 0) will be located inside the circle.
(-3, 3) will be located outside the circle.
(-2, -2) will be located on the circle.
What is the equation of a circle?In Mathematics, the standard form of the equation of a circle is represented by this mathematical expression;
(x - h)² + (y - k)² = r²
Where:
h and k represents the coordinates at the center of a circle.r represents the radius of a circle.In this exercise, we would use an online graphing calculator to plot the equation of this circle as shown in the image attached below.
(x - 2)² + (y - 1)² = 25
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Somebody please help :Write an equation that is perpendicular to x - 3y = 3 and passes through the point (5, -9) in
in with test.
Oy = 3x - 6
Oy=-3x + 6
O y = 1/3 x + 1
Oy=1/3x -1
Answer:
y = - 3x + 6
Step-by-step explanation:
the equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y- intercept )
given
x - 3y = 3 ( subtract x from both sides )
- 3y = - x + 3 ( divide through by - 3 )
y = [tex]\frac{1}{3}[/tex] x - 1 ← in slope- intercept form
with slope m = [tex]\frac{1}{3}[/tex]
given a line with slope m then the slope of a line perpendicular to it is
[tex]m_{perpendicular}[/tex] = - [tex]\frac{1}{m}[/tex] = - [tex]\frac{1}{\frac{1}{3} }[/tex] = - 3 , then
y = - 3x + c ← is the partial equation
to find c substitute (5, - 9 ) into the partial equation
- 9 = - 3(5) + c = - 15 + c ( add 15 to both sides )
6 = c
y = - 3x + 6 ← equation of perpendicular line
. Molly travels 90km to get home after work. She leaves at 6pm and arrives home at 7:30pm.
Calculate her average speed in km per hour
Answer:
60 Kilometers per hour
Step-by-step explanation:
To solve for how long it took Molly to get home, we can use this equation:
7:30 - 6 = 1:30So, Molly took an hour and 30 minutes, or 90 minutes to travel 90km.
To get the number of km she traveled every hour, we can use this equation:
90 × [tex]\frac{2}{3}[/tex] = 60Why do we multiply by 2/3?
We multiply by [tex]\frac{2}{3}[/tex] because if we need to figure out the average speed for every hour (60) we need to convert 90 to 60, and to do that we multiply the 90 by [tex]\frac{2}{3}[/tex].
Therefore, Molly travels 60km every hour.
what is quarts to oz?
The quarts to Oz conversion is simple and can be done through the formula - Quantity in ounces or oz = quantity in quarts × 32
Quart and oz are the units used to get an information on the volume. The two units are related and can be easily converted to each other through the relation. The equivalent amount of one over the other is -
1 quart is equal to 32 fluid ounces.
Now, to find the amount of quarts in oz, students can simply use this formula -
Quantity in ounces or oz = quantity in quarts × amount of ounces equivalent to one quart
The value of amount of ounces equivalent to one quart will be kept 32. So, let us find the 0.03 quart equivalent in ounces.
Quantity in ounces = 0.03 × 32
Performing multiplication on Right Hand Side of the equation
Quantity of fluid in ounces = 0.96 ounces.
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find the slope of the line that passes through the points ( - 1, 2 ) and ( 2, -3 )
1. -5/3
2. 3/5
3. -3/5
4. 5/3
The slope of the line passing through the points (-1, 2) and (2, -3) is -5/3
Calculating the slope of a lineFrom the question, we are to determine the slope of the line that passes through the points (-1, 2) and (2, -3)
To find the slope of the line passing through the points (-1,2) and (2,-3), we can use the formula
m = (y₂ - y₁)/(x₂ - x₁)
Where m is the slope of the line
and (x₁, y₁) and (x₂, y₂) are the coordinates of the two points.
Substituting the given values into this formula, we get
m =(-3 - 2)/(2 - (-1))
m = -5/(2 + 1)
m = -5/3
Hence, the slope is -5/3
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The sum of the first m terms of an arithmetic sequence with first term −5 and common difference 4 is 660. Find m
Based on the arithmetic sequence with the first term −5 and the common difference 4 is 660, we could find m = 20.
To find the sum of the first m terms of an arithmetic sequence, we can use the formula:
sum = (m/2)(2a + (m-1)d),
where a is the first term,
d is the common difference,
and m is the number of terms.
In this case, we are given a = −5, d = 4, and sum = 660. We can plug these values into the formula and solve for m:
660 = (m/2)(2(-5) + (m-1)(4))
660 = (m/2)(-10 + 4m - 4)
660 = (m/2)(4m - 14)
1320 = m(4m - 14)
0 = 4m^2 - 14m - 1320
Now we can use the quadratic formula to solve for m:
m = (-b ± √(b^2 - 4ac))/(2a)
m = (-(−14) ± √((−14)^2 - 4(4)(-1320)))/(2(4))
m = (14 ± √(196 + 21120))/8
m = (14 ± √21316)/8
m = (14 ± 146)/8
Now we have two possible values for m:
m = (14 + 146)/8 = 160/8 = 20
m = (14 - 146)/8 = -132/8 = -16.5
Since m must be a positive integer, we can conclude that m = 20. Therefore, the sum of the first 20 terms of the arithmetic sequence is 660.
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