The arithmetic series 1+5+9+. . . . . +41+45 can be expressed in summation notation as [tex]\sum\limits^{12}_{n=1} {(4n-3)}[/tex]
The given arithmetic series:
1+5+9+. . . . . +41+45
To write the series in summation notation:
Find the common difference (d) of the series.
d = 5 - 1 = 4
Recall the formula for the nth term in an arithmetic sequence:
a(n) = a(1) + (n-1) . d
Substitute a(1) = 1 and d = 4:
a(n) = 1 + (n-1) . 4
a(n) = 4n - 3
Find n that leads to a(n) = 45
a(n) = 4n - 3
45 = 4n - 3
4n = 48
n = 12
Thus, the lower limit of the summation is n =1, and the upper limit is n = 12
Now write in summation notation as:
[tex]\sum\limits^{12}_{n=1} {a(n)}= \sum\limits^{12}_{n=1} {(4n-3)}[/tex]
This summation notation can be further simplified using the properties of summation operator:
[tex]\sum\limits^{12}_{n=1} {(4n-3)}=\sum\limits^{12}_{n=1} {4n}-\sum\limits^{12}_{n=1} {3}[/tex]
[tex]=\sum\limits^{12}_{n=1} {4n}-12\times3[/tex]
[tex]=\sum\limits^{12}_{n=1} {4n}-36[/tex]
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HELP WITH THIS MATH PROBLEM PLS PLS
Answer:
MP = 29
Step-by-step explanation:
Given:
The entire line (MP) is equal to 5y + 9MN = 17NP = 3yMN + NP = MP
17 + 3y = 5y + 9
Find what y =____
17 + 3y = 5y + 9
subtract 9 from both sides
17 - 9 + 3y = 5y + 9 - 9
8 + 3y = 5y
subtract 3y from both sides
8 + 3y - 3y = 5y - 3y
8 = 2y
divided both sides by 2 to get y alone
8/2 = 2y/2
4 = y
Now plug in y to MP to find its length
5(4) + 9 = 20 + 9 = 29
MP = 29
To find NP, plug in y as well
3y = 3(4) = 12
NP = 12
Check:
MN + NP = MP
17 + 12 = 29 yes!!!!
3/2(2c+7) -3c =6(c+5)
Answer:
c = -3 1/4
Step-by-step explanation:
You want the solution to 3/2(2c+7) -3c =6(c+5).
SolutionSimplifying the equation gives ....
3c +21/2 -3c = 6c +30
21/2 -60/2 = 6c . . . . . . . . . subtract 30
-39/2 = 6c . . . . . . . . . . combine terms
-13/4 = c . . . . . . . . . . divide by 6
The value of c is -3 1/4.
please help asap!!!!!!!!!!!!!!!!!!!!!!!
Answer:
MP = NQ = 5 units=====================
Given pointsM = (3, 2), N = (3, -1), P = (7, - 1), Q = (7, 2)Use distance formulaThe distance between points (x₁, y₁) and (x₂, y₂) is:
[tex]d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]Apply this to segments MP and NQ[tex]MP=\sqrt{(7-3)^2+(-1-2)^2}=\sqrt{4^2+(-3)^2}=\sqrt{16+9} =\sqrt{25}=5[/tex][tex]NQ=\sqrt{(7-3)^2+(2-(-1))^2}=\sqrt{4^2+3^2}=\sqrt{16+9} =\sqrt{25}=5[/tex]So we have both segments with same length of 5 units
Complete the following statement.
According to the Angle Bisector Theorem, if a point is on the bisector of an angle, then it is
The point bisector hypothesis expresses that on the off chance that a point is on the bisector of a point, the point is equidistant from the sides of the point.
What is the Angle Bisector Theorem?
In math, the point bisector hypothesis is worried about the overall lengths of the two fragments that a triangle's side is separated into by a line that cuts up the contrary point.
The point bisector hypothesis talk expresses that on the off chance that a point is in the inside of a point and equidistant from the sides, then it lies on the bisector of that point.
According to the angle bisector theorem, PQ/PR = QS/RS or a/b = x/y. A point bisector is a line or beam that separates a point in a triangle into two equivalent measures.
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Tickets for a school event were $1.75 for students and $3.75 for parents. If a total of 62 tickets were sold for $178.50, how many student and parent tickets were sold? There were ___ student tickets sold and ____ parent tickets sold.
Work to solve p and s
Let s = students, s is in the place of x
Let p = parents, p is in the place of y
Answer:
There were 27 student tickets sold and 35 parent tickets sold
Step-by-step explanation:
Use the system of equations by making two equations (one equation for the number of tickets sold and an other for the price the tickets): s + p = 62 and 1.75s + 3.75p = 178.50Cancel out one of the variables by multiplying the first equation of the system (s + p = 62) by its inverse to cancel out; -1.75(s + p = 62) (to eliminate s) or -3.75(s + p = 62) (to eliminate p) to get -1.75s - 1.75p = -108.50 or - 3.75s - 3.75p = -232.50Subtract the first equation from the second equation (to eliminate one variable either s or p) 1.75s + 3.75p = 178.50 - 1.75s - 1.75p - 108.50 that becomes 2p=70 (since both s cancel out) or 1.75s + 3.75p = 178.50 - 3.75s - 3.75p - 232.5 it becomes -2s = -54 (since both p cancel out)Divide both sides by the coefficient of the variable so 2p/2 = 70/2 which is p = 35 or -2s/-2 = -54/-2 which is s = -27 Find the other variable value by substitute your known value and subtracting it on both sides: s + 35 -35 = 62 - 35 this equals to s = 27 or 27- 27 + p = 62 which equals to p = 35
Evaluate each expression for a=3 and b=-5 . -4+2 a b
Evaluating the expression -4 + 2ab for a = 3 and b = -5 is equal to -34.
An algebraic expression is a number, variable, or the combination of both and operational symbols. On the other hand, an equation is the equality of two expressions separated by "=".
To evaluate an expression or an equation means to substitute a value to the variable and perform the arithmetic operations included.
Given the expression -4 + 2ab, substitute the value of the variable a and the value of variable b.
-4 + 2ab
-4 + 2(3)(-5)
Performing the arithmetic operations, we can have the value of the expression.
-4 + 2(3)(-5)
= -4 + (-30)
= -34
Hence, the expression -4 + 2ab when a = 3 and b = -5 is equal to -34.
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Angels (Line). I'm unsure of how to do this.
Answer:
90°
Step-by-step explanation:
X = 90°
A right angle is 90° and they are all 90° (right-angles).
As well as that is, that there are 360° on a point
9x4=36 so it must be the same 90° (1 right angle) times 4
90x4 = 360
Given point M is between L and N, where LM = 10 and NM = 8, What is LN?
Answer:LN=18
Since point M is between the two other points, you add each number, giving you the answer
Task #6
How many buckets do we need?
Abinash wants to fill his pool but his hose is busted! His
friends decided to help him fill the pool with water buckets.
The base of the pool has dimensions 4 m by 2 m and the
height is 1 m, but they decided to fill the pool only
two-thirds way.
Each water bucket has radius
0.13 m and height 0.55 m.
a) How many buckets of water would they need to fill the pool two-thirds way?
b) What assumption did you make to arrive at your answer?
HELPPPPPP! How many groups of 9/2 are in 1?!?
An exchange student is moving back to Italy, and her homeroom class wants to get her a going away present. The teacher takes a survey of the class of 32 students and finds that 10 people chose a card, 12 chose a T-shirt, 6 chose a video, and 4 chose a bracelet. If the teacher randomly selects the present, what is the probability that the exchange student will get a card or a bracelet?
According to the question, the homeroom class wants to get a present for a student. From [tex]32[/tex]students, [tex]10[/tex] choose cards, [tex]12[/tex] choose t-shirts, [tex]6[/tex] choose a video and the remaining choose a bracelet.
Now, calculate the probability that the exchange students should get a card or a bracelet.
Let us assume, ‘C’ represents the card and ‘B’ represents the bracelet.
Using standard probability formula:
P (C or B) = P(C)+P(B) = [tex]\frac{10}{32} + \frac{4}{32} = \frac{14}{32} = \frac{7}{16} = 43.8%[/tex]
Therefore, the probability to get a card or a bracelet is [tex]43.8[/tex] percent.
What is Probability?
The term probability means the study of the occurrence or the outcomes of the event. It is the measurement of the events that occurred during events. And many uncertain events cannot be measured.
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Which quadrilateral is trapezoid?
Answer:
A.
Step-by-step explanation:
A quadrilateral is a trapezoid if two opposite sides are parallel, and the other 2 opposite sides are not parallel.
Answer: A.
Answer:
it's a
Step-by-step explanation:
A die is rolled twice. What is the probability that the first number rolled is a 3 and the second number rolled is a 5 ?
When a die is rolled twice, the probability of getting a 3 and then a 5 is 1/36.
When a die is rolled twice, the following combinations are possible:
1, 1 - 1, 2 - 1, 3 - 1, 4 - 1, 5 - 1, 6
2, 1 - 2, 2 - 2, 3 - 2, 4 - 2, 5 - 2, 6
3, 1 - 3, 2 - 3, 3 - 3, 4 - 3, 5 - 3, 6
4, 1 - 4, 2 - 4, 3 - 4, 4 - 4, 5 - 4, 6
5, 1 - 5, 2 - 5, 3 - 5, 4 - 5, 5 - 5, 6
6, 1 - 6, 2 - 6, 3 - 6, 4 - 6, 5 - 6, 6
Thus, the sample space is 36.
3, 5 only occurs once in the sample space.
∴ P (3 and 5) = 1/36.
Hence, the probability of rolling a 3 and then a 5 is 1/36.
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Each Sophomore and Junior at a high school collected aluminum cans and plastic bottles. The table shows the average number of cans and bottles collected per student by grade level during a 2-week recycling drive.
a. Write a system of equations to represent the situation. Let x represent the number of sophomores, and let y represent the number of juniors. The equation for week 1 is (Type an equation.)
The equation for week 1 at a high school showing collected aluminum cans and plastic bottles is; x + 4y = 2319
How to create Simultaneous Linear Equations?From the given table, we are told that;
Average number of cans and bottles collected by Sophomores in week 1 = 1
Average number of cans and bottles collected by Sophomores in week 2 = 4
Average number of cans and bottles collected by Juniors in week 1 = 4
Average number of cans and bottles collected by Juniors in week 2 = 2
Total number of cans and bottles collected by all in week 1 = 2319
Total number of cans and bottles collected by all in week 2 = 2290
Let x represent the number of sophomores, and let y represent the number of juniors. Thus, the equation for week 1 is;
x + 4y = 2319
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Please i need help, can someone help me pls :<
Answer:
3-2.253.03964-10197Step-by-step explanation:
You want the quadratic expression evaluated for several different values of x.
SolutionEvaluating a polynomial is often easier when it is written in Horner form.
f(x) = -x² -2x +3
f(x) = (-x -2)x +3 = -(x +2)x +3
Repetitive evaluations are often easier to do by programming a spreadsheet or calculator to do them. The basic idea is to put the value where x is, then do the arithmetic.
1. f(0)f(0) = -(0 +2)(0) +3 = 3
2. f(1.5)f(1.5) = -(1.5 +2)(1.5) +3 = -(3.5)(1.5) +3 = -5.25 -3 = -2.25
3. f(-0.02)f(-0.02) = -(-0.02 +2)(-0.02) +3 = (1.98)(0.02) +3 = 3.0396
4. f(-1)f(-1) = -(-1 +2)(-1) +3 = 1 +3 = 4
5. f(100)f(100) = -(100 +2)(100) +3 = -10200 +3 = -10197
__
Additional comment
Horner form rewrites the polynomial to minimize the number of arithmetic operations required to evaluate it.
ax^5 +bx^4 +cx^3 +dx^2 +ex +f = ((((ax +b)x +c)x +d)x +e)x +f
Which of the following is equal to 7^1/4?
A. 4√7
B.√7.4
C. 7^4
D. √7
If quadrilateral WXYZ is reflected across the y -axis to become quadrilateral W' X' Y' Z' , what are the coordinates of X' ?
When the quadrilateral WXYZ is mirrored across the y axis to form the quadrilateral W' X' Y' Z', the coordinates of X' are (-1,6).
Briefing:When the x value is reflected across the y-axis, it takes on the opposite sign.
What exactly is a coordinate system?A coordinate system is a method of determining the position of points or other geometric components on a manifold, such as Euclidean space, by using one or even more numbers, or coordinates.
Why do we employ a coordinate system?A coordinate system is a framework for specifying the relative locations of objects in a specific area, such as an area here on earth's surface or the entire earth's surface. A geographic coordinate system determines locations on the world by using a three-dimensional spherical surface.
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Identify a pattern and find the next three numbers in the pattern. 30,45,60,75,. . .
The next three numbers in the pattern are 80, 85, 90....
These are the multiples of 5.
The complete pattern is 30,45,60,75,80, 85, 90....
What does pattern completion entail?Look at the first three or four numbers in the pattern. You might wonder what link there is between these figures. In this mathematical pattern, for instance, the first three to four numbers are 1, 3, 6, and 10. It seems like you must add to get the higher numbers as you go up the scale. Start with adding 2, then follow with 2, 3, and finally 4. So, if you write this down, you probably see a pattern starting to emerge.
Try the remaining elements of the supplied pattern to see whether your solution works.
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State whether the sentence is true or false. If false, replace the underlined word or number to make a true sentence.
Angles 4 and 8 are corresponding angles.
The statement "∠4 and ∠8 are corresponding angles" is true
Given,
The angles = ∠4 and ∠8
Here, ∠4 and ∠8 are lies same side of the one of two lines and both are same side of the transversal
So, ∠4 and ∠8 are corresponding angles
So the given statement is true
Hence, the statement "∠4 and ∠8 are corresponding angles" is true
Complete question is :
State whether the sentence is true or false. If false, replace the underlined word or number to make a true sentence.
∠4 and ∠8 are corresponding angles.
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At your job, you get paid 1.5 times your hourly wage for any hours worked over 40 hours. You work 52 hours during a week in the summer. Your total paycheck was for $982.50. What is your hourly wage?
At the job, the hourly Wage was solved to be $16.94.
How to find the hourly wage
Given data
rate of hours worked over 40 hours = 1.5
hours worked in a week = 52
total paycheck = $982.50
let the pay hourly rate be x, and over 40 hours pay is 1.5x
finding the hours worked over 40
52 - 40 = 12
total paycheck $982.50 is gotten as follows:
$982.50 = 40 * x + 12 * 1.5x
982.50 = 40x + 18x
982.50 = 58x
x = 982.50 / 58
x = 16.9397
x = 16.94
Since x which is the hourly wage is found to be 16.94m, we can therefore say that the hourly wage is $16.94.
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168>=7x+4y
i need help solving for y
btw this is an inequality
For solving y, we can use this inequality: y≤(168-7x)/4
What is an inequality and steps to solve an inequality ?An inequality is an equation which compares two non equal expressions using operators ≤,≥,<,> .
Steps for solving inequalities:
if we add same number to both sides the inequality remains same.If we divide both sides by a positive number the inequality remains same .if we divide both sides by a negative number the inequality , the inequality reverses.For given inequality, 168≥ 7x + 4y
Adding -7x both sides,
168-7x ≥ 7x+4y-7x
168 - 7x ≥ 4y
Dividing both sides by a positive number 4,
(168-7x)/4 ≥ y
we can solve for y using above expression that y will be less than or equal to (168-7x)/4
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MP.7 Use Structure Sue and Jonah
chose numbers for a place-value game.
Sue chose the number one hundred
fifty-two thousand. Jonah chose five
million for his number. Who chose the
greater number? Explain.
Jonah has chosen the greater number.
Given, Sue and Jonah chose numbers for a place value game.
As a way to express numbers, a number system is a way of expressing them. It is a method of describing numbers from a particular set mathematically by consistently employing digits or other symbols. Every number is given a distinct representation, and the figures' algebraic and mathematical structures are also shown.
Sue chose the number = one hundred fifty two thousand
= 152000
= 1.52 lakhs
Jonah chose the number = five million
= 5,000,000
we know that 10 lakhs = 1 million = 1,000,000
therefore, 5 million = 5 × 10 lakhs
= 50 lakhs
50 lakhs > 1.5 lakhs
the greater number is 5 million.
Therefore , the greater number chose by Sue and Jonah is of Jonah's that is 5 million.
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Evaluate the indicated function for f(x) = x2 − 1 and g(x) = x − 3 algebraically. If possible, use a graphing utility to verify your answer. (fg)(4t2)
STEP 1: Apply the definition of the product of two functions.
(fg)(4t2) =
STEP 2: Replace the f and g expressions in your answer to Step 1 with expressions involving only t.
(fg)(4t2) =
STEP 3: Expand your answer from Step 2.
(fg)(4t2) =
The value of the indicated function would be (fg)(4t2) = (64t⁶ - 48t⁴ - 4t² + 3)
How to solve for the functionThe question has that
f(x) = x² - 1
f(g) = x -3
We have to find (fg)(4t²)
(fg)(4t²) = f(4t²) (g(4t²)
= (4t²)²-1) (4t² - 3)
we have to expand the bracket
(16t⁴ - 1) (4t² - 3)
= (64t⁶ - 48t⁴ - 4t² + 3)
Hence we can say that the value of
(fg)(4t2) = (64t⁶ - 48t⁴ - 4t² + 3)
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Find the 32 nd term of each sequence. 0.1,0.5,0.9,1.3, ...........
The 32nd term of the arithmetic sequence AP is computed to be 12.5.
What is an AP arithmetic sequence?In mathematics, an arithmetic progression (AP) is a list or sequence of numbers in which each term is attained by adding a pattern to the term before it.
With in arithmetic progression's common difference, the fixed number is marked by the letter 'd.'The common difference => d = a2 - a1 = a3 - a2 = a4 - a3 =...... = a - an-1.nth term of an AP: an = a + (n - 1) dThe sum of n terms of an AP's => Sn = n/2(2a+(n-1)d) = n/2(a + l), where l is the last term of the AP.Finally, here is the answer to the question;
The following are the numbers in the presented sequence:
0.1,0.5,0.9,1.3, ...........
Because the series contains 32 terms, we must find the 32nd number.
Assume the first term is'a₁' = 0.1.
Assume 'a₂' = 0.5 is the second term.
Assume the third term is 'a₃' = 0.9
'd' is the common difference between two consecutive terms of an AP that are equal.
d = a₂ - a₁
Put the values in the preceding formula; the 32nd term is
d = 0.5 - 0.1
d = 0.4
The nth term equation is found by the formula;
nth term of an AP: an = a + (n - 1) dTotal number of terms is; n = 32Initial term is a = 0.1Common difference d = 0.4Substitute the given values in the formula;
a₃₂ = a + (n - 1) d
a₃₂ = 0.1 + (32 - 1)(0.4)
a₃₂ = 0.1 + 12.8 - 0.4
a₃₂ = 12.5
Therefore, the arithmetic sequence's 32nd term value is estimated to be 12.5.
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find the midpoint of the segment that has endpoints (-3,4) and (-6,0)
Answer:
the mid point of (-3,4) and (-6,0) is (4.5,2)
Answer:
[tex]=(-4.5, 2)[/tex]
Step-by-step explanation:
Midpoint formula is:
[tex](x_m, y_m) = (\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})[/tex]
where
x1=-3, x2=-6
y1=4, y2=0
Now just plug into formula
[tex]=( \frac{-3+-6}{2}, \frac{4+0}{2})[/tex]
[tex]= ( \frac{-9}{2}, \frac{4}{2})[/tex]
[tex]=(-4.5, 2)[/tex]
How many binary strings of length 10 are there in which the number of 0's in the string is not equal to the number of 1's in the string?
The binary strings of length 10 are there in which the number of 0's in the string is not equal to the number of 1's in the string is 2¹⁰ - 2⁹.
Binary number:
A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method of mathematical expression which uses only two symbols: typically "0" and "1".
Given,
Here we need to find How many binary strings of length 10 are there in which the number of 0's in the string is not equal to the number of 1's in the string.
Here the length of the binary string is 10.
So, it can be written in the power form as 2¹⁰.
Now we need to find the number of 0's in the string is not equal to the number of 1's in the string.
For that, we have to consider,
There are 2⁹ number of binary strings are not have same number of 0's in the string is not equal to the number of 1's in the string.
So, it can be written as,
=> 2¹⁰ - 2⁹
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which inequality matches this graph?
Answer:
x>4
Step-by-step explanation:
Since it is pointing right, it must be either
x>4
x≥4
Since it is not a filled circle the answer can't be greater than or equal to. So it must be x>4.
Do 300 MM, 190 MM make a triangle?
The side lengths 300 mm and 190 mm make a valid triangle because of the triangle inequality theorem
How to determine if the side lengths make a triangle?The side lengths are given as
300 mm and 190 mm
To solve this question, we make use of the triangle inequality theorem
The triangle inequality theorem states that
Given that the sides of a triangle are x, y and z.
The sum of any two sides of a triangle is greater than or equal to the third side;
This means that
x + y ≥ z
Using the above as a guide, we have the following inequality expressions
300 + 190 ≥ z
300 + z ≥ 190
z + 190 ≥ 300
Evaluate the inequality expressions
490 ≥ z
z ≥ -110
z ≥ 110
Remove the negative value
490 ≥ z
z ≥ 110
Combine the inequalities
110 ≤ z ≤ 490
The above inequality is a valid inequality expression
Hence, the side lengths 300 mm and 190 mm make a valid triangle
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Mr. Sanchez’s class sold fruit pies for $1.89 each and Mr. Kelly’s class sold bottles of fruit juice for $1.56 each. Together, the classes sold 82 items and earned $140.13 for their school. Write and solve a system of equations that models this problem. Show all your work!
Mr. Sanchez's class sold fruit pies, and Mr. Kelly's class sold bottles of fruit juice.
Mr. Sanchez's class sold 45 fruit pies, and Mr. Kelly's class sold 37 bottles of fruit juice.
What are linear equations?Linear equations are equations that have constant average rates of change. Note that the constant average rates of change can also be regarded as the slope or the gradient
How to write and solve a system of equations that models this problem.To solve this question, we use the following representations:
x represents the fruit pies
y represents the fruit juice
A system of linear equations is a collection of at least two linear equations.
In this case, the system of equations is given as
1.89x + 1.56y = 140.13 ---- the total cost
x + y = 82 --- the number of sales
Make x the subject in x + y = 82
So, we have
x = 82 - y
Substitute x = 82 - y in 1.89x + 1.56y = 140.13
1.89(82 - y) + 1.56y = 140.13
Expand
154.98 - 1.89y + 1.56y = 140.13
Evaluate the like terms
-0.33y = -14.85
Divide by -0.33
y = 45
So, we have
x = 82 - y
x = 82 - 45
Evaluate
x = 37
Hence, Mr. Sanchez's class sold 45 fruit pies, and Mr. Kelly's class sold 37 bottles of fruit juice.
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16. The denominator of a fraction is 4 more than the numerator. If 1 is added to numerator and 2 is subtracted from denominator fraction becomes 24 upon 25 find the fraction
The fraction is 23/27.
Let the numerator and denominator of the given fraction be x and y respectively.
According to the question,
Case 1 :-
Denominator of the fraction is 4 more than the numerator.
⇒ Numerator = Denominator - 4
⇒ x = y - 4 ...(1)
Case 2 :-
If 1 is added to numerator and 2 is subtracted from denominator fraction becomes 24/25
⇒ (Numerator + 1) / (Denominator + 2) = 4/5
⇒ (x + 1) / (y - 2) = 24/25
⇒ 25(x + 1) = 24(y - 2)
⇒ 25x + 25 = 24y - 48
⇒ 25x - 24y = -73 ...(2)
Substituting the value of x from (1) in (2), we have
⇒ 25(y - 4) - 24y = -73
⇒ 25y - 100 - 24y = -73
⇒ y = -73 + 100
y = 27
substitute the value of y in equation 1.
x = 27 - 4
x = 23
Now, At the beginning of our solution we took the numerator & denominator to be x & y respectively. So the fraction is:
⇒ x / y
⇒ 23/27
Hence the fraction is 23/27.
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