3/2 x 4/3 x 5/4… x 2006/2005
Answers
Answer:
1003
Step-by-step explanation:
The problem is a classic example of a telescoping series of products, a series in which each term is represented in a certain form such that the multiplication of most of the terms results in a massive cancelation of subsequent terms within the numerators and denominators of the series.
The simplest form of a telescoping product\(a_{k} \ = \ \displaystyle\frac{t_{k}}{t_{k+1}}\), in which the products of n terms is
\(a_{1} \ \times \ a_{2} \ \times \ a_{3} \ \times \ \cdots \times \ a_{n-1} \ \times \ a_{n} \ = \ \displaystyle\frac{t_{1}}{t_{2}} \ \times \ \displaystyle\frac{t_{2}}{t_{3}} \ \times \ \displaystyle\frac{t_{3}}{t_{4}} \ \times \ \cdots \ \times \ \displaystyle\frac{t_{n-1}}{t_{n}} \ \times \ \displaystyle\frac{t_{n}}{t_{n+1}} \\ \\ \-\hspace{5.55cm} = \ \displaystyle\frac{t_{1}}{t_{n+1}}.\).
In this particular case, \(t_{1} \ = \ 2\) , \(t_{2} \ = \ 3\), \(t_{3} \ = \ 4\), ..... , in which each term follows a recursive formula of \(t_{n+1} \ = \ t_{n} \ + \ 1\). Therefore,
\(\displaystyle\frac{t_{2}}{t_{1}} \times \displaystyle\frac{t_{3}}{t_{2}} \times \displaystyle\frac{t_{4}}{t_{3}} \times \cdots \times \displaystyle\frac{t_{n}}{t_{n-1}} \times \displaystyle\frac{t_{n+1}}{t_{n}} \ = \ \displaystyle\frac{3}{2} \times \displaystyle\frac{4}{3} \times \displaystyle\frac{5}{4} \times \cdots \times \displaystyle\frac{2005}{2004} \times \displaystyle\frac{2006}{2005} \\ \\ \-\hspace{5.95cm} = \ \displaystyle\frac{2006}{2} \\ \\ \-\hspace{5.95cm} = 1003\)
Related Questions
Using the formula :-
\(\tt A\:=\:PMT\:\cdot \left[\frac{\left(1+\cfrac{APR}{n}\right)^{nY}-1}{\cfrac{APR}{n}}\right]\)
Find the savings plan balance after 2 years with an APR of 6% and monthly payments of $150.
got my answer as 3814.79, just want to check my work, thanks!:)
Answers
The savings plan balance after 2 years with an APR of 6% and monthly payments of $150 is approximately $3,814.79.
What are payments?
In compensation for products or services received or to fulfill a legal duty, payment is the voluntarily given exchange of money, its equivalent, or other valuables from one person to another. Frequently, the individual giving the money is referred to as the payer, while the person receiving it is referred to as the payee.
Assuming that the payments are made at the end of each month, we can use the formula -
\(A = PMT\Bigg[\frac{\big(1+\frac{APR}{n}\big)^{nY}-1}{\frac{APR}{n}}\Bigg]\)
where -
A = the savings plan balance after 2 years
PMT = the monthly payment ($150)
APR = the annual percentage rate (6%)
n = the number of compounding periods per year (12, since the payments are made monthly)
Y = the number of years (2)
Plugging in the values, we get -
\(A = 150\Bigg[\frac{\big(1+\frac{0.06}{12}\big)^{12 \times 2}-1}{\frac{0.06}{12}}\Bigg]\)
A ≈ $3,814.79
Therefore, value is obtained as $3,814.79.
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is the ordered pair a solution to the equation. y=-7; (-5,6)
Answers
Answer: No
To check if (-5,6) is a solution to the equation y=-7, we need to substitute -5 for x and 6 for y in the equation.
If the equation is true, then the ordered pair is a solution to the equation.
If we put (-5,6) into y=-7, we get 6=-7, which is not true.
So, (-5,6) is not a solution to the equation y=-7.
Let f(x)=x+1 and g(x)=5x evaluate the composition (f•g)(-1)
Answers
Answer:
\((f\circ g )(x) = -4\)
Step-by-step explanation:
We have
\((f\circ g )(x) := f(g(x))\)
Therefore,
\(f(x)=x+1 \implies f(x)=g(x)+1\)
So,
\((f\circ g )(x) = 5x + 1\)
For \(x = -1:\)
\((f\circ g )(x) = 5(-1) + 1 = -5+1=-4\)
Solve for p -14 = -(p - 8)
Answers
This is simple algebra. The process of solving for p is shown below:
\(\begin{gathered} p-14=-\mleft(p-8\mright) \\ p-14=-p+8 \\ p+p=8+14 \\ 2p=22 \\ p=\frac{22}{2} \\ p=11 \end{gathered}\)p is 11
I need helps this is an assignment dealing with kites
Answers
In a kite, there is one pair of congurent angles. So
\(3x-22=x+52\)Solve the equation for x.
\(\begin{gathered} 3x-22=x+52 \\ 3x-x=52+22 \\ x=\frac{74}{2} \\ =37 \end{gathered}\)So value of x is 37.
Answer: 37
Help please! Look at pic ASAP! Giving brainlest to the right amswer
Answers
Answer:
27 divided by 6=?
=4.5
divide the 27 into 6 equal container to get the amount of ounce per container
How to simplify 3/18 it’s a fraction and it’s really hard can someone go step by step how to do it I need it ASAP
Answers
Answer:1/6
Step-by-step explanation: divide both sides by 3 as 3 itself and 18 can be divided by 3
Two cars are traveling towards a hotel on the same road. From the edge of the hotel, 600 feet high, Spiderman sits on the rooftop thinking about the depression angle needed to reach each car. If the depression angle to the nearest car is 52 degrees, and the depression angle to the farther car is 46 degrees, how far apart must the two cars be from each other?
Make a sketch, solve the problem, and round your answer to the nearest hundredth of a foot.
Answers
The two cars must be approximately 177.34 feet apart from each other for Spiderman to have different depression angles to each car.
To find the distance between the two cars, we can use trigonometry and the concept of similar triangles. Let's denote the distance between Spiderman and the nearest car as d1 and the distance between Spiderman and the farther car as d2.
In a right triangle formed by Spiderman, the height of the hotel, and the line of sight to the nearest car, the tangent of the depression angle (52 degrees) can be used:
tan(52) = 600 / d1
Rearranging the equation to solve for d1:
d1 = 600 / tan(52)
Similarly, in the right triangle formed by Spiderman, the height of the hotel, and the line of sight to the farther car, the tangent of the depression angle (46 degrees) can be used:
tan(46) = 600 / d2
Rearranging the equation to solve for d2:
d2 = 600 / tan(46)
Using a calculator, we can compute:
d1 ≈ 504.61 feet
d2 ≈ 681.95 feet
The distance between the two cars is the difference between d2 and d1:
Distance = d2 - d1
Plugging in the values, we have:
Distance ≈ 681.95 - 504.61
Distance ≈ 177.34 feet
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This may be hard to put together, but I need help with an angle equation. I'm looking for example equations and how to do these types of equations. *please note that this equation is completely random and probably ends with an ugly number. it's just an example of a type of problem*
Answers
Example: (2x - 4) x=7 so (2x (x=7) - 4) =18
Fill in the table using this function rule y= - 2x + 3
Answers
Step-by-step explanation:
you just have to put that value in place of x in the function.
don't be so lazy you a hole do it yourself
find the mean, median, mode, range, standard deviation, and interquartile range.
Answers
The statistical measures for the frequency table are given as follows:
Mean: 65.Median: 10.5.Mode: Class 9 - 12.Range: 6.Standard deviation: 1.96.IQR = 3.5.How to obtain the statistical measures?Considering the midpoints of the middle intervals, the frequencies are given as follows:
Age of 8: Frequency of 1.Age of 10.5: Frequency of 7.Age of 14: Frequency of 5.The mean is given by the sum of all observations divided by the number of observations, hence:
Mean = (8 + 7 x 10.5 + 5 x 14)/13 = 11.65.
The median is the middle element of the data-set. The data-set has the 13 elements, hence the median is the 7th element, which is in the class range 9-12, hence it is of 10.5.
The mode is the class with the most observations, hence it is the class 9 -12.
The range is the difference between the highest value and the lowest, hence:
14 - 8 = 6.
The standard deviation is the square root of the sum of the differences squared between each observation and the mean, hence:
S = square root((1 x (8 - 11.65)² + 7 x (10.5 - 11.65)² + 5 x (14 - 11.65)²)/13) = 1.96.
The quartiles are given as follows:
First quartile -> median of the first six elements -> 10.5.Third quartile -> median of the last six elements -> 14.Then the IQR is calculated as follows:
IQR = 14 - 10.5 = 3.5.
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Find the length of the third side. If necessary, round to the nearest tenth. 8 12
Answers
The length of the third side is 14.4 units.
What is Trigonometry?Trigonometry is a branch of mathematics that studies relationships between side lengths and angles of triangles.
We find the third side of a triangle by using Pythagoras theorem.
In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides
8²+12²=x²
64+144=x²
208=x²
Take square root on both sides
x=14.4
Hence, the length of the third side is 14.4 units.
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What is the solution for x^2+ 4x > 77?
x<-7 or x > 11
x<-11 or x > 7
-7
-11
Answers
Answer:
x<-11 or x>7
Step-by-step explanation:
First, we get the right side to be 0.
\(x^2+4x-77 > 0\)
Then we factor the left side.
\((x+11)(x-7) > 0\)
Using the Snake Method, we determine the intervals as x<-11 or x>7
3(y-5)= 15 please give me the answer!!!!
Answers
Answer: y=10
Step-by-step explanation: Divide both sides by 3, then add 5 to both sides. 15/3=5, 5+5=10.
Answer:
y = 10
Step-by-step explanation:
3(y-5) = 15
y - 5 = 5
y = 5 + 5 = 10
Identify the graph of the inequality 2(2x-1)+7< 13 or -2x+5-10.
Answers
From the resulting solution, the correct linear inequality graph is Graph C.
Solving inequality expressionGiven the inequality equation below:
2(2x-1)+7< 13 or -2x+5 ≤ -10.
Simplify the expression
2(2x-1)+7< 13
Expand
4x - 2 + 7 < 13
4x + 5 < 13
4x < 13 - 5
4x < 8
x < 2
For the inequality -2x+5 ≤ -10.
-2x+5 ≤ -10
-2x ≤ -15
x ≥ 7.5
Hence the solution to the given system of inequalities are x < 2 and x ≥ 7.5
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Rotation of 270° (x,y) or (a,b) becomes
Answers
A totation of 270º will chncge the sign of the x and y coordinate so the condinates will change:
\(\begin{gathered} (x,y)\to(-x,-y) \\ (a,b)\to(-a,-b) \end{gathered}\)Use distributive property to evaluate each expression
A.5(x-12)=
B. Y(12+8)=
C.3(x+y)=
Answers
20y
3x+3y
Thats it lol
B.20y
C.3x+3y
hope this helps
A hardware store spent $13,290 on print and TV advertising last year.
2
If of that amount was spent on print advertising, how much was
spent on TV advertising?
Select one:
O a. $5,316
O b. $13,290
O c. $18,606
C.
O d. $7,974
Answers
So the correct answer is $6,645, or option D. If just a portion of that sum was spent on print advertising.
What is division?Multiplication is the inverse of division. If 3 groups of 4 add up to 12, 12 split into 3 equal groups adds up to 4 in each group in division. The basic purpose of splitting is to determine how many equal groups develop or how many people are in each group when distributing equally. A division in mathematics is the process of dividing a specified amount into equal pieces. For example, we can split a group of 20 people into four groups of five people each, or five groups of four people each, and so on.
Here,
If half of the $13,290 was spent on print advertising, then $13,290 / 2 = $6,645 was spent on print advertising.
Therefore, $13,290 - $6,645 = $6,645 was spent on TV advertising.
So, the answer is $6,645, or option D If of that amount was spent on print advertising.
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Question:-
The area of two similar triangles are 81 cm2 and 49 cm² respectively. If one of the sides of the first triangle is 6.3 cm, find the corresponding side of the other triangle.
Answers
Let's assume that the corresponding side of the second triangle is \(\displaystyle\sf x\).
The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. Therefore, we can set up the following proportion:
\(\displaystyle\sf \left( \dfrac{x}{6.3} \right)^{2} =\dfrac{49}{81}\)
To find \(\displaystyle\sf x\), we can solve the proportion above:
\(\displaystyle\sf \left( \dfrac{x}{6.3} \right)^{2} =\dfrac{49}{81}\)
Taking the square root of both sides:
\(\displaystyle\sf \dfrac{x}{6.3} =\sqrt{\dfrac{49}{81}}\)
Simplifying the square root:
\(\displaystyle\sf \dfrac{x}{6.3} =\dfrac{7}{9}\)
Cross-multiplying:
\(\displaystyle\sf 9x = 6.3 \times 7\)
Dividing both sides by 9:
\(\displaystyle\sf x = \dfrac{6.3 \times 7}{9}\)
Calculating the value:
\(\displaystyle\sf x = 4.9\)
Therefore, the corresponding side of the second triangle is \(\displaystyle\sf 4.9 \, cm\).
\(\huge{\mathfrak{\colorbox{black}{\textcolor{lime}{I\:hope\:this\:helps\:!\:\:}}}}\)
♥️ \(\large{\underline{\textcolor{red}{\mathcal{SUMIT\:\:ROY\:\:(:\:\:}}}}\)
Answer:
Step-by-step explanation:
let's assume that the corresponding side of the second triangle is .
The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. Therefore, we can set up the following proportion:
To find , we can solve the proportion above:
Taking the square root of both sides:
Simplifying the square root:
Cross-multiplying:
Dividing both sides by 9:
Calculating the value:
Therefore, the corresponding side of the second triangle is 4.9cm
hope it helped u dear...........
Evaluate x^{3} - 3y when x = -3 and y = 4.
Answers
Answer:
-39
Step-by-step explanation:
Substitute x by - 3 and y by 4
(-3)^3 - 3 *4 = - 39
evaluate f(-x)=x/x2+1
Answers
To evaluate f(-x) when f(x) = x/(x^2 + 1), we need to substitute -x for x in the function.
So f(-x) = (-x)/((-x)^2 + 1)
Simplifying the denominator we get:
f(-x) = (-x)/(x^2 + 1)
The function f(-x) is equivalent to the function f(x) with the x-value negated. The value of the function f(-x) is the same as the value of the function f(x) but with the x-value negated.
It's important to note that we can also express this as f(-x) = -f(x)
So the value of f(-x) is x/(x^2 + 1) with the x-value negated.
Ordered pair for 2x+3y=5x-y
Answers
True or false? In a two-column proof, the left column states your reasons.
A. True
B. False
Answers
A. True.
In a two-column proof, the left column consists of statements, which are the facts or assumptions that lead to the proof of the theorem, while the right column consists of the reasons or justifications that explain how each statement logically follows from the previous one. Therefore, the left column states your reasons is a true statement.
The answer to the student's question is True. In a two-column proof, the left column contains the statements (steps) and the right column contains the corresponding reasons (justifications).
Explanation:The answer to the question, 'True or false? In a two-column proof, the left column states your reasons', is A. True.
A two-column proof is organized into two columns. The left column contains the 'Statements' and the right column contains the corresponding 'Reasons'. The 'Statements' are the steps that lead to the conclusion of the proof, while the 'Reasons' justify each of those steps according to rules or laws of mathematics.
For example, if we want to prove that the opposite angles of a parallelogram are equal. The left column (Statements) could contain the first step 'ABCD is a parallelogram' and the right column (Reasons) would give the explanation 'Given'. Therefore, in a two-column proof, the left column does represent statements, not reasons.
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How many inches is in 5 1/4 feet
Answers
Answer:
63 in.
Step-by-step explanation:
Answer:
There is 63 inches in 5 1/4 feet :)
How to solve it&&&&&&&&&&&
Answers
Answer:
y = 4 and x = 12
Step-by-step explanation:
Step by step explanation in the pic. Atleast the way I did it.
What are the next numbers in the series 1, 2, 3, 0, 1, 2, -1
Answers
The next numbers in the series will be 0,1,-2,-1,0,-3,-2,-1........
What is number Series?This refers to a pattern of arranging numbers. In the number series problems, we will have to make the rules that result in the formation of a number series.
From the question:
1, 2, 3, 0, 1, 2, -1
The pattern is:First value plus one, second value plus one, Third value minus three. Then the pattern starts afresh again.
Calculating the pattern we have:1 +1 = 2
2 + 1 = 3
3 -3 =0
0+1 = 1
1 + 1 = 2
2 -3 =-1
-1 + 1 =0
0 + 1 =1
1 - 3 = -2
-2 + 1 = -1
-1 + 1 = 0
0 -3 = -3
-3 + 1 = -2
-2 + 1= -1
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Alisa uses her cell phone to search how many pounds of sand will is required to fill 1 cubic foot and finds out the answer is 100 pounds. Choose 1 of the brands and compute how much it will cost Alisa to purchase enough sand to fill the court. Sand A - 12 dollars per 150 lb, Sand B - 5 dollars per 60 pounds. BRO PLEASE HELP, THIS THING IS DUE IN A WEEK AND IM ABSOLUTELY CLUELESS
Answers
Using proportions, the cheapest option for Alisa to purchase enough sand to fill the court is Option A, which will cost $8.
What is a proportion?
A proportion is a fraction of a total amount, and the measures are related using a rule of three. Due to this, relations between variables, either direct or inverse proportional, can be built to find the desired measures in the problem.
For each option, the cost per pound is
Option A: 12/150 = $0.08.Option B: 5/60 = $0.0833.Option A is cheaper, hence the total cost for 100 pounds is:
$0.08 x 100 = $8.
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The height of cylinder B is twice the height of cylinder A the total surface area of cylinder A is 180 Calculate the total surface area of cylinder B
Answers
Main Answer: The total surface area of cylinder B is approximately 476.7 square units.
Supporting Question and Answer:
How is the surface area of a cylinder calculated?
The surface area of a cylinder is the sum of the areas of its curved surface and two circular bases. It can be calculated using the formula:
A = 2πrh + 2π\(r^{2}\)
where "r" is the radius of the circular base, "h" is the height of the cylinder, and π (pi) is a mathematical constant that represents the ratio of the circumference of a circle to its diameter, approximately equal to 3.14159.
The first term, 2πrh, represents the area of the curved surface of the cylinder, while the second term, 2π\(r^{2}\), represents the combined area of the top and bottom circular bases.
Body of the Solution: Let the height and radius of cylinder A be "h" and "r" respectively. Then, the height and radius of cylinder B are "2h" and "R" respectively, since cylinder B has twice the height of cylinder A but the same radius.
The total surface area of a cylinder is given by the formula:
Area = 2πrh + 2π\(r^{2}\)
For cylinder A, we know that the surface area is 180, so we can substitute the values and solve for "r" as follows:
180 = 2πrh + 2π\(r^{2}\)
90 = πrh + π\(r^{2}\)
We can rearrange this equation to solve for r:
r² + rh - 90/π=0
Using the quAdratic formula,we get:
r = (-h ± √((h)² + 4(90/π)))/2
r = -h/2 ± √(h² + 4(90/π))/2
Now we have an expression for r in terms of h . We can use this expression to find the radius of cylinder B, since we know that cylinder B has twice the height of cylinder A.
R = 2r= -h ± √(h² + 4(90/π))
We can use this expression to find the surface area of cylinder B:
Surface area of cylinder B = 2πR² + 2πR(2h)
Surface area of cylinder B= 2π( -h ± √(h² + 4(90/π)))² + 4πh( -h± √(h² + 4(90/π)))
We can use the value we found for 2h using the quadratic formula in the expression to get:
Surface area of cylinder B = 476.7 square units (rounded to one decimal place)
Therefore, the total surface area of cylinder B is approximately 476.7 square units.
Final Answer: Therefore, the total surface area of cylinder B is approximately 476.7 square units.
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what's the answer
1. The function below shows the weekly earnings of Alice from her job last summer.
Answers
a) We kindly invite to check the image attached below to see a detailed graph of the piecewise function.
b) The piecewise function is continuous.
How to understand a piecewise functionA piecewise function is a conditional combination of two or more functions, whose expression depends on which value of the domain is the function evaluated at.
a) The piecewise function described in the question is plotted on a graphing tool (i.e. Desmos), whose result is presented in the image attached below.
b) A function is continuous if and only if there is one value from range for every value of domain. The piecewise function is continuous as linear functions are continuous and the two functions of the conditional combination have the same value for x = 30.
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What is the measure of the other leg of this triangle?
Answers
Answer:
48.2593.3424
Step-by-step explanation:
40+.2 and then you complete it by saying.35times 65