The sum of the first five terms of an arithmetic sequence is 35. The sum of the next five terms is 85. Determine the first term and the common difference of the sequence.
Answer:
The first term is 3 and the common difference is 2
Step-by-step explanation:
The formula of the sum of the nth term of the arithmetic sequence is
[tex]S_{n}=\frac{n}{2}[2a+(n-1)d][/tex], where
a is the first termd is the common differencen is the position of the term∵ The sum of the first five terms of an arithmetic sequence is 35
∴ [tex]S_{5}[/tex] = 35
→ Substitute The values of n and S in the rule above
∵ 35 = [tex]\frac{5}{2}[2a+(5-1)d][/tex]
∴ 35 = [tex]\frac{5}{2}[2a+4d][/tex]
→ Simplify the right side
∴ 35 = 5a + 10d
→ Switch the two sides
∴ 5a + 10d = 35 ⇒ (1)
∵ The sum of the next five terms is 85
→ To find the sum of the first 10 terms add 85 to [tex]S_{5}[/tex]
∴ [tex]S_{10}[/tex] = 35 + 85 = 120
→ Substitute The values of n and S in the rule above
∵ 120 = [tex]\frac{10}{2}[2a+(10-1)d][/tex]
∴ 120 = 5[2a + 9d]
→ Simplify the right side
∴ 120 = 10a + 45d
→ Switch the two sides
∴ 10a + 45d = 120 ⇒ (2)
Now we have a system of equations to solve it and find a and d
→ Multiply equation (1) by -2
∵ -2(5a) + -2(10d) = -2(35)
∴ -10a + -20d = -70 ⇒ (3)
→ Add equations (2) and (3)
∵ (10a + -10a) + (45d + -20d) = (120 + -70)
∴ 25d = 50
→ Divide both sides by 25 to find d
∴ d = 2
→ To find a substitute d in equation (1) by 2
∵ 5a + 10(2) = 35
∴ 5a + 20 = 35
→ Subtract 20 from both sides
∵ 5a + 20 - 20 = 35 - 20
∴ 5a = 15
→ Divide both sides by 5
∴ a = 3
∴ The first term is 3 and the common difference is 2
Need help :((
plss help i took a screenshot so you can see
Answer: 5:4
Step-by-step explanation: you divide them i believe :)
Answer:
5:9
Step-by-step explanation:
Well it says the ratio to the blueberry muffins to all muffins so add together 12+15 which is 27 then compare the ratios in simplest form so 15:27 in simplified form is 5:9
A loaf of bread costs $4.00 today. The same size loaf cost 20 cents in 1955. What percentage of today's price did someone pay for bread. Explain your thinking.
Answer:
$0.8, or 80 cents
Answer:
Your answer is 0.05 or5%
Step-by-step explanation:
what you do is 20÷400=0.05=5%
What is f(0) if f(t) = t^2 – 2t – 2?
Answer:
-2
Step-by-step explanation:
f(t) = t^2 – 2t – 2
To find f(0), replace t with 0 in the expression and evaluate the expression.
f(0) = 0^2 – 2(0) – 2
f(0) = -2
0 = t
We know this because 0 replaced t in the parenthesis next to F !
The question is just asking you to plug in 0 where ever t is!
Let's do it
[tex]f(0) = 0^{2} - 2(0) - 2[/tex]
solve
[tex]0^{2} = 0 \\2 ( 0 ) = 0[/tex]
so, this is what we have left!
[tex]f(0) = 0 - 0 - 2[/tex]
We have our answer!
[tex]f (0) = -2[/tex]
Which graphs display a directly proportional relationship? Check all that apply. 2
Answer:The first picture, second picture,fourth picture represents proportional relationship.
Further explanation:
Explanation:
The points in the first picture are and
The slopes between the points can be obtained as follows,
The slopes between the points are equal. Therefore, in the first picture x and y are in a proportional relationship.
The points in the second picture are and
The slopes between the points can be obtained as follows,
The slopes between the points are equal. Therefore, in the second picture x and y are in a proportional relationship.
The points in the third picture are and
The slopes between the points can be obtained as follows,
The slope is negative. Therefore, in the third picture x and y are not in a proportional relationship.
The points in the fourth picture are and
The slopes between the points can be obtained as follows,
The slopes between the points are equal. Therefore, in the fourth picture x and y are in a proportional relationship.
The first picture, second picture,fourth picture represents proportional relationship.
Learn more:
Learn more about inverse of the function brainly.com/question/1632445.
Learn more about equation of circle brainly.com/question/1506955.
Learn more about range and domain of the function brainly.com/question/3412497
Answer details:
Grade: High School
Subject: Mathematics
Chapter: Ratio and proportion
Keywords: function, First picture, second picture, proportional relationship, ratio, slope, set, set of values, set of numbers, coordinates, x-coordinate, y-coordinate.
Step-by-step explanation:
Fill in the blank. I NEED AN ANSWER FASTTT
Answer:
It is =
Step-by-step explanation:
Because (6 x 1 = 6 ) + 1 x 1/100 = 6.1
Solve the following absolute value equation'
2|x+3|-2=0
A(-2,-4)
B(4,2)
C(-3)
D(1,-7)
Answer: A(-2,-4)
Explanation:
2|x+3|-2 = 0
2|(-2)+3|-2 = 0
2|(-4)+3|-2 = 0
If you take a number multiply by 7 then takeaway 1,You get the same as if you took the number multiply by 5 then takeaway 8. What is the number?
Answer:
x = -3.5
Step-by-step explanation:
First we create the equation.
Let x = the unknown number
7x-1 = 5x - 8
Add 8 to both sides
7x + 7 = 5x
Subtract 7x
7 = -2x
Divide by 2
-7/2 = x
-3.5 = x
The slide at a playground
has a height of 22 feet. The
distance between the foot
of the ladder and the end of
the slide is 10 feet. What is
the length of the slide?
Round to the nearest tenth.
Answer:
c = 24.17 feet
Step-by-step explanation:
Right Triangles
A right triangle is a special triangle that has an interior angle of 90°.
In right triangles, Pythagora's Theorem is satisfied:
[tex]c^2=a^2+b^2[/tex]
Where c is the hypotenuse (longest side) of the triangle, and a and b are the legs or the smaller sides.
The slide can be thought of as a right triangle whose hypotenuse is the slide itself and the legs are the height of the ladder and distance between the foot of the ladder and the end of the slide.
So we have:
[tex]c^2=22^2+10^2[/tex]
[tex]c^2=484+100=584[/tex]
[tex]c=\sqrt{584}[/tex]
c = 24.17 feet
Whats The Answer Marking Brainliest, Please explain your answer:D..............................Please HELP!!!
It’s not an question per say, I wanted to see if my answer was right
Answer:
Yes it’s correct I did this for a test like last week lol
Step-by-step explanation:
Answer:
YEs
Step-by-step explanation:
a+650=<800
a=>150
yes it is correct
Hope this helps plz hit the crown ;D
Which best describes the effect on the x-intercept of the graph of y=34x−3 if the slope is changed to −34
The x-intercept remains the same and the new line is translated upwards
The x-intercept becomes negative and the new line is parallel to the original line
The x-intercept remains the same and the new line is translated downward
The x-intercept becomes negative and the new line intersects the original line
Answer:
The last option: The x-intercept becomes negative and the new line intersects the original line.
Step-by-step explanation:
Remember that when finding the x-intercept you set y to 0.
Let's compare the two equations when we solve for x:
0=34x-3 0=-34x-3
1. Add 3 to both sides of the equations
3=34x 3=-34x
2. Divide the coefficients
[tex]\frac{3}{34}[/tex]=x [tex]\frac{3}{-34}[/tex]=x
We know that the new slope does not remain the same as stated in the first and third options. In order for two lines to be parallel, the slopes must be exactly the same, so the second option is incorrect. Therefore, the last option is correct.
Answer:
Step-by-step explanation:
⊕ The x-intercept becomes negative and the new line intersects the original line { at (0, - 3) }
In the equation 5 + 3x - 7 = 10, what are the like terms?
Answer:
5 , -7 , 10
Step-by-step explanation:
You are buying a coke for yourself and tacos for your friends. You have a limited budget of y dollars. If the coke cost $2.25 and the tacos cost $1.25 each, write a linear inequality to help you determine how many tacos you can buy.
Answer:
2.25x+1.25y[tex]\leq[/tex]y
Step-by-step explanation:
Step one:
given data
the budget of y dollars
the coke cost $2.25 and
the tacos cost $1.25
Required:
the linear inequality based on the limited budget
Step two:
let the number of coke be x
and the number of tacos be y
So the linear inequality is
2.25x+1.25y[tex]\leq[/tex]y
Answer:
The answer is 2.25x+1.25yy
Step-by-step explanation:
area of a rectangle whose length is 4x-6 and width is x+1
Answer: The area is 4x^2-2x-6
Step-by-step explanation: The area of a rectangle is length*width. So multiply both these terms. Remember the method foil, that would apply here. (4x-6)(x+1), you end up with 4x^2+4x-6x-6. Then combine like terms, to get 4x^2-2x-6.
The graph of y = f(x) is the solid black graph below. Which function represents the dotted graph?
Answer:
y = -f(x - 5)
Step-by-step explanation:
the function of the dotted graph is f(x) shifted (or translated) 5 units to the right and then reflected across the x-axis.
shifting it to the right is happening by reducing every x value by 5 (so that we get for x the functional value of x-5).
and reflecting it across the x-axis simply means to flip the sign. everything that was originally above the x-axis is now with the same distance below the x-axis (and vice versa).
therefore, the new
y = -f(x - 5)
HELP this is important!
Answer:
1. 11,17,23,29
2.t=19
3. +6
4. 4
5.57
Which situation is best represented by the following equation?
45w+123.95=753.95
A.Erica paid $753.95 for dance classes. She paid $123.95 for each week she was enrolled in the classes after using a coupon that gave her $45 off the price per week. What is w,the number of weeks Erica was enrolled in dance classes?
B.Erica paid $753.95 for dance classes. She paid a $123.95 registration fee and $45 for each week she was enrolled in the classes. What is w, the number of weeks Erica was enrolled in dance classes?
C.Erica paid $753.95 for dance classes. She paid a $45 registration fee and $123.95 for each week she was enrolled in the classes. What is w, the number of weeks Erica was enrolled in dance classes?
D.Erica and her sister paid $753.95 for dance classes. Erica paid $123.95 for each week she was enrolled in the classes, and her sister paid $45 for each week she was enrolled in the classes. What is w, the number of weeks Erica and her sister were enrolled in dance classes?
Answer:
The correct answers is B.
Step-by-step explanation:
I need help please!!!
Answer:(y=2x+2
Step-by-step explanation:
You decide to put a new stone siding on the rectangular side of your
house, which measures 8 by 15 feet. If stone costs $3 per square foot,
what is the total cost of stone?
Which point below has a positive x-value and a negative y-value? Q RS on
What is the slope of the line that passes through points (6, − 2) and (21, 3) ?
Answer: __________
Answer:
[tex]m=\frac{1}{3}[/tex]
General Formulas and Concepts:
Pre-Algebra
Order of Operations: BPEMDAS
Brackets Parenthesis Exponents Multiplication Division Addition Subtraction Left to RightAlgebra I
Slope Formula: [tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]Step-by-step explanation:
Step 1: Define
Point (6, -2)
Point (21, 3)
Step 2: Find slope m
Substitute: [tex]m=\frac{3+2}{21-6}[/tex]Add/Subtract: [tex]m=\frac{5}{15}[/tex]Simplify: [tex]m=\frac{1}{3}[/tex]What is the equation of this line in slope intercept forn?
3z +y=2
A. y=-3+2
B. y=3z +2
C. y=-3z 2
D. y=3r-2
Answer:
We conclude that the equation of this line in the slope-intercept form will be:
[tex]y = -3z+2[/tex]Step-by-step explanation:
We know that the slope-intercept form of the line equation is
[tex]y=mx+b[/tex]
where m is the slope and b is the y-intercept.
Given the equation
[tex]3z +y=2[/tex]
Writing the equation in the slope-intercept form of the line equation by simplifying it
subtracting 3z from both sides
[tex]3z+y-3z = -3z+2[/tex]
[tex]y = -3z+2[/tex]
Therefore, we conclude that the equation of this line in the slope-intercept form will be:
[tex]y = -3z+2[/tex]Lauren saved $25 a month for 2 years. She spent 75% of her savings on a TV. How much money does Lauren have left?
Answer
$150
Step-by-step explanation:
Since there are 12 months in year, you multiply 25 times 24 to get 600. 75% of 600 is 450, so then you do 600-450. The answer to that is 150.
Hope this helped!
Barbara bought a piece of rope that was 6 3/4 meters long. She cut the rope into 6 equal pieces. How long is each piece of rope?
Answer:
1 1/8
Step-by-step explanation:
Make this an improper fraction for simplicity: 27/4
now divide into six portions: (27/4)/6 = (27/6)/4 = 27/24 = 1 3/24 = 1 1/8
12-e <-2
solution???
12 - e < -2
- e < -2 - 12 Move 12 to subtract.
- e < - 14
e > 14 When the coefficient is in negative. Moving the negative as changing the sign of Inequality to opposite.
Therefore, the answer is e > 14
You take a taxi to your sisters home. The driver charges you $8 for 3 kilometers and $2 for each additional kilometer. If you paid a total of $14 how long did you travel
Kate and Nora each have a sum of money. The ratio of the amount of money Kate has to that of Nora is 3 : 5. After Nora gives $150 to Kate, the ratio of the amount of money Kate has to that of Nora becomes 7 : 9. Find the sum of money Kate had initially.
Given:
The ratio of the amount of money Kate has to that of Nora is 3 : 5.
After Nora gives $150 to Kate, the ratio of the amount of money Kate has to that of Nora becomes 7 : 9.
To find:
The sum of money Kate had initially.
Solution:
Let the initial amount of money Kate and Nora have are 3x and 5x respectively.
After Nora gives $150 to Kate, the ratio of the amount of money Kate has to that of Nora becomes 7 : 9.
The amount of Kate = 3x+150
The amount of Nora = 5x-150
Now,
[tex]\dfrac{3x+150}{5x-150}=\dfrac{7}{9}[/tex]
[tex]9(3x+150)=7(5x-150)[/tex]
[tex]27x+1350=35x-1050[/tex]
[tex]1350+1050=35x-27x[/tex]
[tex]2400=8x[/tex]
Divide both sides by 8.
[tex]300=x[/tex]
Putting x=300 in 3x, to get the amount of money Kate has initially.
[tex]3(300)=900[/tex]
Therefore, the Kate initially had $900.
What is the area of a circle with a 17cm diameter?
Please help! i will give brainliest.