Try adding up the first few odd numbers and see if the numbers you get satisfy some sort of pattern. Once you find the pattern, express it as a formula. Give a geometric verification that your formula is correct.
The required sum of the first n odd numbers is equal to n².
How we take numbers to make pattern?The first few odd numbers are 1, 3, 5, 7, 9, 11, and so on. When we add up the first few of these numbers, we get the following sums:
1 = 1
1 + 3 = 4
1 + 3 + 5 = 9
1 + 3 + 5 + 7 = 16
1 + 3 + 5 + 7 + 9 = 25
We can see that the nth odd number is equal to 2n - 1. And the sum of the first n odd numbers is equal to n². This can be verified geometrically by considering a square with sides of length n.
The sum of the first n odd numbers can be thought of as the sum of the diagonal elements of the square, which forms a square of length n.
Thus, the sum of the first n odd numbers is equal to n².
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PLEASE HELP!! MATH PROBLEM!!
please answer and explain
Classify each function as a power function, root function, polynomial (state its degree), rational function, algebraic function, trigonometric function, exponential function, or logarithmic function.
(a) y=Ï€x
(b) y=xπ
(c) y=x2(2−x3)
(d) y=tant−cost
(e) y=s1+s
(f) y=x3−1√1+x√3
(a) f(x) = [tex]log_2(x)[/tex] is a logarithmic function
(b) g(x) = ∜x is a root function
(c) h(x) = [tex]2x^3/(1 - x^2)[/tex] is a rational function
(d) u(t) = 1 - 1.1t + [tex]2.54t^2[/tex] is a polynomial of degree 2
(e) v(t) = [tex]5^t[/tex]is an exponential function
(f) w(θ) = sin θ [tex]cos^{2}\theta[/tex] is a trigonometric function.
(a) f(x) = [tex]log_2(x)[/tex] is a logarithmic function. Logarithmic functions have the logarithm of the independent variable as the output. Here, the logarithm base is 2.
(b) g(x) = ∜x is a root function. Root functions have the square root or higher roots of the independent variable as the output. Here, the root is a cube root.
(c) h(x) = [tex]2x^3/(1 - x^2)[/tex] is a rational function. Rational functions are functions that are expressed as the quotient of two polynomials. Here, the numerator is a cubic polynomial and the denominator is a quadratic polynomial.
(d) u(t) = 1 - 1.1t + [tex]2.54t^2[/tex] is a polynomial of degree 2. Polynomials are functions that are expressed as a sum of powers of the independent variable, with coefficients. The degree of a polynomial is the highest power of the independent variable.
(e) v(t) = [tex]5^t[/tex] is an exponential function. Exponential functions have the independent variable as the exponent. Here, the base is 5.
(f) w(θ) = sin θ [tex]cos^{2}\theta[/tex] is an algebraic function and a trigonometric function. Algebraic functions are functions that can be expressed using arithmetic operations and algebraic expressions. Trigonometric functions are functions that involve the ratios of the sides of a right triangle. Here, the function is a combination of sine and cosine functions.
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The complete question is -
Classify each function as a power function, root function, polynomial (state its degree), rational function, algebraic function, trigonometric function, exponential function, or logarithmic function.
(a) f(x) = [tex]log_2(x)[/tex]
(b) g(x) = ∜x
(c) h(x) = [tex]2x^3/(1 - x^2)[/tex]
(d) u(t) = 1 - 1.1t + [tex]2.54t^2[/tex]
(e) v(t) = [tex]5^t[/tex]
(f) w(θ) = sin θ [tex]cos^{2}\theta[/tex]
Lacey pays $15 for each hour of golf lessons and $10 for equipment rental. If the lesson is more than 3 hours long, she only pays $5 for equipment rental. Is the total cost of Lacey’s lesson a function of the number of hours scheduled? Use a table to help explain your answer. Show your work.
Answer:
60
Step-by-step explanation:
15x3=45
5x3=15
15+45=60
FIND MULTIPLICITY AND ZEROS OF THE GRAPH!
Frank is going to plant b vegetable seeds in one garden and 5b + 6 vegetable seeds in another . How many seeds is frank going to plant ?
The total number of seeds Frank is going to plant is 6b + 6 seeds.
How many seeds is frank going to plant ?From the question, we have the following parameters that can be used in our computation:
Graden 1 = b
Graden 2 = 5b + 6
Frank is going to plant b seeds in one garden and 5b + 6 seeds in another garden.
The total number of seeds Frank is going to plant is:
b + (5b + 6) = 6b + 6
So Frank is going to plant 6b + 6 seeds.
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Select the correct answer from each drop-down menu.
In a sequence described by a function, the notation f(2) = 4 means that the
term of the sequence has a value of
. Sequences can be described as functions because
.
The function that describes the sequence maps each index or position to its corresponding value.
What does "sequence" signify in functional analysis?A sequence space is a vector space that contains endless sequences of real or complex numbers. It is used in functional analysis and related mathematical fields. It is, in essence, a function space whose constituents are functions from the natural numbers to the field K of real or complex numbers.
The notation f(2) = 4 indicates that a term in a sequence defined by a function has a value of 4 when the term's index or position is 2.
Because sequences are a collection of ordered pairs with the first element being the index or position of the term and the second being the term's value, they can be characterised as functions. The function that represents the series maps each index or location to its associated value, and these ordered pairs can be seen as points on a coordinate plane.
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I don’t think 53/63 is in lowest terms haha so help is def needed
Answer:
53/63
Step-by-step explanation:
It is already in simplest form
To find the average speed of an object, we find the change in distance divided by the time over which that change occurs. For the instantaneous speed, we must make the time interval very _______, so that the speed does not change appreciably.
To find the instantaneous speed of an object, we must make the time interval very small, so that the speed does not change appreciably.
How to find the speed?
Speed is a scalar quantity which tells how fast an object is moving, regardless of its direction.
To find the instantaneous speed of an object, we must make the time interval very short, so that the speed does not change appreciably. The formula for the instantaneous speed of an object is given by the limit as time approaches zero of the change in distance divided by the time over which that change occurs. The formula for average speed is simply the total distance traveled divided by the total time elapsed.
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If x= 45, then √x is between
4 and 5
44 and 46
22 and 23
6 and 7
Help me pleaseeeeeeeee
Answer:
61
Step-by-step explanation:
√(2+3)^2+(2+4)^2
=√5^2+6^2
=√25+36
=√61
So the number is 61
3. Mr. Jackson could think about his experiment a little differently. How many times should he expect to roll a 4 if he rolls his number cube 30 times? This is called the mean, or expected value. You can find it if you know the number of trials and the probability of success for an individual trial. Use the formula μx = np to find the expected value.
The probability of having a 4 in a 30 roll is 5 times
What is the probability distribution
The expected value, μ, of rolling a 4 on a fair number cube (with faces numbered 1 to 6) can be found by using the formula μ = np, where n is the number of trials (30 in this case) and p is the probability of success for an individual trial (rolling a 4).
The probability of rolling a 4 on a fair number cube is 1/6. Therefore, the expected value of rolling a 4 30 times can be calculated as follows:
μ = 30 * (1/6) = 5
So, Mr. Jackson can expect to roll a 4 approximately 5 times in 30 rolls of his number cube.
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Find f’(x) for f(x)=4x^3-5x^2+7x-1
The scale of the map is missing. The actual distance from Granada Hills to Malibu is 54 miles, and it is 9 inches on the map. What is the scale of the map?
y= -3x+7 = what number is for solving y?
Answer:
y=-3x+7
its the same thing
Step-by-step explanation:
I need help with these questions ?!
Answer:
Step-by-step explanation:
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Answer:
The first one is 3/5 & The second one is 3/8 & The the third one is 1/8
Step-by-step explanation:
Hope this helps :)
A new mixture of self - tanning & moisturizing lotions for everyday use is being developed . This mixture is made by mixing 700 ounces of everyday moisturizing lotion for $0.70 per ounce with self - tanning lotion worth $2 per ounce. If the new self - tanning & moisturizing lotion mixture is to cost $ 1.30 per ounce, how many ounces of the self-tanning lotion should be in the mixture?
The mixture should have ( ) ounces of the self - tanning lotion.
Last night, the Dallas Mavs made 12 free throws. If
their free throw percentage is 75%, how many free
throws were attempted?
Question 19). TPRS is a parallelogram. Identify m∠R.
Solve what is in the attatchment!!
the measurement of the ∠R will be 101 degrees.
What is a parallelogram?A quadrilateral with the opposing sides parallel is called a parallelogram (and therefore opposite angles equal). A parallelogram with all right angles is known as a rectangle, while a quadrilateral with equal sides is known as a rhombus.
Given TPRS is a parallelogram. in that ∠T = (4x +1) and ∠S = (3x +4)°
Since in a parallelogram, opposite angles are equal Thus,
∠P = ∠S = 3x + 4
∠R = ∠T = 4x +1
Since we know the sum of all angles of a quadrilateral is 360°
3x +4 + 4x +1 + 3x +4 + 4x +1 = 360
14x + 10 = 360
14x = 350
x = 350/14
x = 25
Thus,
∠R = 4x +1
∠R = 4 *25 + 1
∠R = 101°
therefore, the measurement of the ∠R will be 101 degrees.
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The transpose of a product of matrices equals the product of their transposes in the same order. Choose the correct answer below: A.The statement is true. Matrix multiplication is not commutative so the product of the transposes must be applied in the same order as the initial product. B.The statement is true The product of the transposes can be applied in any order. C.The statement is false, The transpose of product of matrices equals the sum of the transposes of the matrices. D.The statement is false. The transpose of a product of matrices equals the product of their transposes in the reverse order
The statement is true. Matrix multiplication is not commutative, so the product of the transposes of the matrices must be applied in the same order as the initial product, not in a reverse order.
The statement that the transpose of a product of matrices equals the product of their transposes in the same order is true. Matrix multiplication is not commutative, which means that the order in which the matrices are multiplied matters. That is, for two matrices A and B, AB does not necessarily equal BA. Thus, when taking the transpose of a product of matrices, the order in which the transposes of the matrices are multiplied must be the same as the order in which the original matrices were multiplied. Otherwise, the result will not be the same. For example, if two matrices A and B are multiplied to give AB, the transpose of AB will be (AB)^T = B^T A^T, not A^T B^T. This is why the product of the transposes must be applied in the same order as the initial product.
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The given pie-chart represents the division of sports played by students during summer camp.
If the total number of students took part in the summer camp was 300
, then how many students played cricket more than basketball?
17 students played cricket more than basketball.
What is a Pie Chart?A pie chart is a type of graph where a circle is divided into certain sectors where each sector shows a proportion of the whole.
Pie chart showing the preference of the students is given below.
Total number of students = 300
We know that total degree of the circle = 360°
Students who played cricket = 90°
Number of students who played cricket will be (90° / 360°)th of 300.
Number of students who played cricket = (90° / 360°) × 300
= 75
Students who played basketball = 70°
Number of students who played basketball = (70° / 360°) × 300
= 58.33 ≈ 58
Students played cricket - Students played basketball = 75 - 58 = 17
Hence there are 17 students more in cricket than that in basketball.
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A tree casts a shadow that is 180 feet long. At the same time, a person standing
nearby who is 5 feet 10 inches tall casts a shadow that is 60 inches long. How tall
is the tree?
Answer: 154.28 feet
Step-by-step explanation:
The mean daily production of a herd of cows is assumed to be normally distributed with a mean of 38 liters, and standard deviation of 4.3 liters.
A) What is the probability that daily production is between 28 and 45.9 liters? Do not round until you get your your final answer.
The probability that daily production is between 28 and 45.9 liters is 0.024.
What is standard deviation?Standard deviation is the positive square root of the variance. Standard deviation is one of the basic methods of statistical analysis. Standard deviation is commonly abbreviated as SD and denoted by 'σ’ and it tells about the value that how much it has deviated from the mean value.
Given that, x₁=28, x₂=45.9, μ=38 and σ =4.3
We know that, Z=(x-μ)/σ
Z=standard score, x=observed value, μ=mean of the sample and σ=standard deviation of the sample
z =(38-28)/4.3= 2.32 and probability on the z-table is 0.9896
z = (38-45.9)/4.3 = -1.83 and probability on the z-table is 0.9656,
Now, P(2.32<z<-1.83) = 0.9896-0.9656
= 0.024
Therefore, the probability that daily production is between 28 and 45.9 liters is 0.024.
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What is an angle that is adjacent to ∠CED?
The angle that is adjacent to the angle CED is the angle DEA.
What are adjacent angles?When two angles have a similar vertex and side, they are referred to as neighbouring angles. The vertex of an angle is the point at which the rays that make up its sides come to an end. When adjacent angles have a same vertex and side, they can be a complimentary angle or supplemental angle.
An adjacent angle is complementary to another if the total of its adjacent angles is 90 degrees.
When the total of two adjacent angles is 180 degrees, they are supplementary to one another.
Two angles are said to be adjacent angles when they share a common vertex.
In the given figure the angle that shares the common vertex to angle CED is angle DEA.
Hence, the angle that is adjacent to the angle CED is the angle DEA.
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The polygon in the diagram is a square with center P. The length from the center to the vertex is 6sqrt(2) in. Find the area of the square to the nearest tenth of an inch.
A) 24.0 in^2
B) 72.0 in^2
C) 36.0 in^2
D) 144.0 in^2
The local zoo nurses 15 species of cats. Last week, several species of cats were moved to other zoos due to lack of funding. The zoo now nurses only 7
species.
Let x represent the number of cats.
Which equation models the scenario?
x 7 = 15
x + 7 = 15
15 + x = 7
15-x = 7
The equation 15 - x = 7 models the scenario because it represents the decrease in the number of cats that the zoo now nurses. It can be read as "15 (the number of cats originally) minus x (the number of cats moved) equals 7 (the number of cats the zoo now nurses)".
Please help I will give brainliest and 5 stars!!
A. The properties of parallelogram involving slope is the slope of opposite sides of parallelogram are equal as the lines are parallel. Hence, the vertices ABCD form a parallelogram.
What is a parallelogram?A quadrilateral with two sets of parallel sides is referred to as a parallelogram. In a parallelogram, the opposing sides are of equal length, and the opposing angles are of equal size. Additionally, the interior angles that are additional to the transversal on the same side. 360 degrees is the sum of all interior angles.
A parallelepiped is a three-dimensional shape with parallelogram-shaped faces.
A. The properties of parallelogram involving slope is the slope of opposite sides of parallelogram are equal as the lines are parallel.
B. The opposite sides of the parallelogram are parallel and equal, and their slopes are equal.
C. The given vertices of a parallelogram are:
A(-2, 0), B(0, 9), C(2, 0) and D(0, -9).
Calculate the slope of segment AB and CD.
The slope is given as:
m = (y2 - y1) / (x2 - x1)
For AB:
m = (9 - 0) / (0 + 2) = 9/2
For CD:
m = (-9 - 0) / (0 - 2) = 9/2
Since the slopes of the segments AB and CD are equal the two lines are parallel to each other. Hence, the vertices ABCD form a parallelogram.
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Nanuk has a savings account which gathers compound interest each year.
He opened the account with a deposit of
£2000.
After 4 years there is £2545.10 in the
account.
Work out the annual interest rate of the
savings account.
Give your answer as a percentage to 1
dp
[tex]~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+\frac{r}{n}\right)^{nt} \quad \begin{cases} A=\textit{accumulated amount}\dotfill & \pounds 2545.10\\ P=\textit{original amount deposited}\dotfill &\pounds 2000\\ r=rate\to r\%\to \frac{r}{100}\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{each year, thus once} \end{array}\dotfill &1\\ t=years\dotfill &4 \end{cases}[/tex]
[tex]2545.10 = 2000\left(1+\frac{\frac{r}{100}}{1}\right)^{1\cdot 4} \implies \cfrac{2545.10}{2000}=\left( 1+\cfrac{r}{100} \right)^4 \\\\\\ \cfrac{2545.10}{2000}=\left( \cfrac{100+r}{100} \right)^4\implies \sqrt[4]{\cfrac{2545.10}{2000}}=\cfrac{100+r}{100} \\\\\\ 100\sqrt[4]{\cfrac{2545.10}{2000}}=100+r\implies 100\sqrt[4]{\cfrac{2545.10}{2000}}-100=r\implies \boxed{6.2\approx r}[/tex]
A newly hired lawyer receives a $15,000 signing bonus from a law firm and invests the money in a savings account at 4.75% interest. After 42 months, the lawyer checks the account balance. Part A: Calculate the interest earned, to the nearest dollar, if the interest is compounded quarterly. Show all work. (2 points) Part B: Calculate the interest earned, to the nearest dollar, if the interest is compounded continuously. Show all work. (2 points) Part C: Using the values from Part A and Part B, compare the interest earned for each account by finding the difference in the amount of interest earned. (1 point)
The amount when compounded quaterly and compounded continuously will be $16,979.44 and $17,713.02. And the difference is $733.58
What is continuous compounding?Theoretically, long-term average interest means that interest is continuously earned on a current account as well as reinvested into the balance to increase future interest earnings.
The equation is given as,
[tex]\rm P=P_o \times e^{rt}[/tex]
The initital amount is $15,000 and the interest rate is 4.75% annually.
A. If the interest is compounded quarterly. Then the amount is given as,
[tex]\rm A = \$15,000 \times \left (1 + \dfrac{0.0475}{4} \right)^{42/4}[/tex]
A = $15,000 x 1.1319
A = $16,979.44
B. if the interest is compounded continuously. Then the amount is given as,
[tex]\rm A = \$ 15,000 \times e^{0.0475 \times (42 / 12)}[/tex]
A = $15,000 x 1.1808
A = $17,713.02
C. The difference in the amount of interest earned is given as,
Differenece = $17,713.02 - $16,979.44
Differenece = $733.58
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Three girls Share an amount of money. The eldest get b/3c of the total amount while the youngest gets gets b/6c of the total, what fraction does the third girl gets?
The fraction does the third girl gets is 3c - b/3c
What is fraction?A fraction can be defined as the part of a whole number or element.
The different types of fraction are;
Proper fractionsImproper fractionsComplex fractionsSimple fractionsMixed fractionsFrom the information given, we have that;
3 girls shares an amount of money
The eldest = b/3x
The youngest = b/6c
This is represented as;
1 - b/3c + b/6c
Add the fractions
1 - 2b - b /6c
add the values
1 - 3b/6c
1 - b/3c
3c - b/3c
Hence, the expression is 3c - b/3c
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