Find the sum of the first 7 terms of the following series, to the nearest integer.
Answer:
32, 40, 50, ...

The special product formula for the 'square of a sum' is

(a + b)² = a² + 2ab + b².

(a + b)² = (a +b)×(a + b)

By applying Distributive law in multiplication, we know

(a + b)² = a×(a + b) + b×(a + b)

(a + b)² = a² + ab + ba + b²

(a + b)² = a² + 2ab + b²

For example, (3 + 2)²

We know that (3 + 2)² = 5² = 25

But let us try applying the special product formula

(3 + 2)² = 3² + 2×3×2 + 2²

(3 + 2)² = 9 + 12 + 4

(3 + 2)² = 25

So we can use this formula in cases where we don't know the square of a sum. Many applications can be seen like solving quadratic equations.

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The values of x for the given quadratic equation 2x² = 4x - 7

are given by:

[tex]x = \frac{2 +i\sqrt{10} }{2}[/tex]

[tex]x = \frac{2 -i\sqrt{10} }{2}[/tex]

What is the solution of a quadratic equation?
The solutions to the quadratic equation are the values of the unknown variable x, which satisfy the equation. These solutions are called roots or zeros of quadratic equations.

According to the given question:

For this case we have the following quadratic equation:

2x² = 4x - 7

Rewriting the equation we have:

2x² - 4x + 7 = 0

From here, we have:

a = 2b = -4c = 7

Substituting values in the quadratic equation we have:

x = (-b ± √b² - 4ac)/2a

x = (-(-4) ± √(-4) - 4(2)(7)/2(2)

Rewriting the equation we have:

x = (4 ± √16 - 56)/4

x = (4 ± √-40)/4

x = (2 ± i√10)/2

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Without using calculator, the values are;

i. sec x = √5/2

ii. cosec x =  -√5

iii. cot x = -2

From the information given, we have that the trigonometric identity tangent takes the value;

tan x = - 1/2

Where;

Now, let's determine the hypotenuse side,

Hypotenuse square = opposite square + adjacent square

substitute the values

x² = (-1)² + (2)²

find the squares

x² = 1 + 4

x = √5

For secant,

sec x = hypotenuse/adjacent

sec x = √5/2

For cosecant,

cosec x = hypotenuse/opposite

cosec x = √5/-1 = -√5

For cot x,

cot x = Adjacent/opposite

cot x = 2/-1

cot c = -2

Hence, the values are √5/2, -√5 and -2

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Answer:

It's A

Step-by-step explanation:

The first part is her purchase before the tax, which should be $32. The second part is her purchase after sales tax, which adds 8% to the shoes and results in the purchase being $34.40

Answer:

ƒ(6) = − 50

Step-by-step explanation:

ƒ(x) = − x² – 14, Find f(6)

ƒ(6) = − 6² – 14

ƒ(6) = − 36 – 14

ƒ(6) = − 50

The value of the function (f /g)(x) would be [tex](\frac{f}{g})(x)=x^2+6[/tex].

Option (A) is correct.

What are composite functions?

A composite function is generally a function that is written inside another function. The composition of a function is done by substituting one function into another function. For example, f [g (x)] is the composite function of f (x) and g (x). The composite function f [g (x)] is read as “f of g of x”.'

Given:

                [tex]f(x)=7x^3-5x^2+42x-30\\\\g(x)=7x-5[/tex]

Now by using the definition of composite functions we can write

      [tex](\frac{f}{g})(x)=\frac{f(x)}{g(x)}\\\\(\frac{f}{g})(x)=\frac{7x^3-5x^2+42x-30}{7x-5}\\\\(\frac{f}{g})(x)=x^2+\frac{42x-30}{7x-5}\\\\(\frac{f}{g})(x)=x^2+6[/tex]

Hence, the value of the function (f - g)(x) would be [tex](\frac{f}{g})(x)=x^2+6[/tex].

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A spinner simulation with an 8-section spinner and 7 trials would be appropriate.

Vince makes 75% of his free throws, so there is a 75% chance that he will make any given free throw. In order to simulate his free throw shooting and predict the probability that he will make 6 of his next 7 free throws, a spinner simulation with an 8-section spinner would be appropriate. The 8 sections would represent the possible outcomes of each free throw attempt: make or miss. The spinner would need to be spun 7 times to represent the 7 free throw attempts Vince will take. This simulation would allow Vince to see the probability of making 6 out of 7 free throw attempts based on his 75% success rate.

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The solution to the system of equations, y = 7x - 3 and 7x - 6y = 18, is: (0, -3).

Given the system of equations:

y = 7x - 3 --> eqn. 1

7x - 6y = 18 --> eqn. 2

Using the substitution method, substitute y for (7x - 3) into eqn. 2:

7x - 6(7x - 3) = 18

7x - 42x + 18 = 18

-35x + 18 = 18

-35x = 18 - 18

-35x = 0

x = 0

To find the value of y, substitute x = 0 into eqn. 1:

y = 7(0) - 3

y = 0 - 3

y = -3

The solution is: (0, -3).

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The median for the next batch of observations is 5.

The median is the value in the middle of a data set, which indicates that 50% of the data points have a value less than or equal to the median, and 50% have a value greater than or equal to the median. The median is the number in the middle of a set of numbers. The median represents the 50th percentile of the set of numbers. In other words, the median is the value in the center of a set of numbers where half are less than the median and half are more than the median.

Here,

given set of observation,

=3,4,5,8,9

n=5

median=(5+1)/2

=3rd term

=5

The median for following set of observation is 5.

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Answer:

250 square feet

Step-by-step explanation:

1 1/2 = 3/2 = 1.5

1.5 pound bag = 375 square feet

Take 375 divided by 1.5 = 250 square feet

So, 1 pound of wildflower seeds will cover 250 square feet!

The relationship between the volume of a cone and a pyramid:

the volume of cone = volume of pyramid

if and only if the pyramid has circular base and both have equal radii and vertical heights

In this question we need to find the relationship between the volume of a cone and a pyramid.

We know that a pyramid is three dimensional shape with polygon base and trigular lateral faces which meet at vertex.

The formula for the volume of pyramid is,

V = 1/3 * base area * height

As we know a cone is nothing but a pyramid with circular base.

Consider a cone with radius r and vertical height h.

The formula for the volume of cone,

V = 1/3 × π × r² × h

V = 1/3 * area of circle * height

Therefore, the volume of a cone and a pyramid with circular base are equal.

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Step-by-step explanation:

The general quadratic equation [tex]ax^2+2hxy+by^2+2gx+2fy+c=0[/tex] represents pair of straight lines if its discriminant [tex]\Delta=0[/tex] and [tex]h^2-ab\geq0.[/tex] The discriminant of this equation is given by the determinant,

[tex]\Delta=\left|\begin{array}{ccc}a&f&g\\f&b&h\\g&h&c\end{array}\right|[/tex]

Here the given equation is,

[tex]\longrightarrow x^2+2xy+2y^2+2x+2y+1=0[/tex]

So,

[tex]a=1[/tex]

[tex]b=2[/tex]

[tex]c=1[/tex]

[tex]f=1[/tex]

[tex]g=1[/tex]

[tex]h=1[/tex]

Thus,

[tex]\longrightarrow\Delta=\left|\begin{array}{ccc}1&1&1\\1&2&1\\1&1&1\end{array}\right|[/tex]

Performing the operation [tex]R_3\to R_3-R_1[/tex] over this determinant,

[tex]\longrightarrow\Delta=\left|\begin{array}{ccc}1&1&1\\1&2&1\\0&0&0\end{array}\right|[/tex]

Now the 3rd row is completely so the discriminant is equal to zero.

[tex]\longrightarrow\Delta=0[/tex]

So our equation represents degenerate conics, which are,

pair of distinct straight lines intersecting each other at any point if [tex]h^2-ab > 0.[/tex]

pair of straight lines, either coincident or distinct but parallel to each other, if [tex]h^2-ab=0.[/tex]

a single point if [tex]h^2-ab < 0.[/tex]

We see that,

[tex]\longrightarrow h^2-ab=1^2-1\times 2=-1[/tex]

[tex]\longrightarrow h^2-ab < 0[/tex]

So our equation represents a single point.

Hence the equation does not represent pair of straight lines.

For a sample population of size, n =5, sx = 20 and sx² = 120, the sum of squared deviation is 40.

Therefore, the answer is 40.

Given a sample population of 5 scores, that is n = 5.

sx = ∑x = 20

sx² = ∑x² = 120

Thus mean can be calculated by

mean, x⁻ = ∑x/ n

x⁻ = 20/ 5

x⁻ = 4

The sum of squared deviations

ss = ∑( x - x⁻)²

ss = ∑( x² -2×x×x⁻ + x⁻²)

ss = ∑x² -2×x⁻×∑x + ∑x⁻²

ss = ∑x² -2×x⁻×∑x + nx⁻²

ss = 120 - 2×4×20 + 5×4²

ss = 40

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1440 combinations ways.

We know that the total arrangements will be

6!= 720 because its a 6letter word

So if we take MI to be a one letter word

Same for IM

which will Also be 6!= 720

SO TOTAL possible combination will be 720+720= 1440

Combinations are a way to calculate the total outcomes of an event where order of the outcomes does not matter. To calculate combinations, we will use the formula nCr = n! / r! * (n - r)!, where n represents the total number of items, and r represents the number of items being chosen at a time.

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Answer:

2 + (S ÷ 180)

Step-by-step explanation:

S is the sum of the measure of the interior angle measures for a polygon, while n is the number of sides.

We are given the equation S = 180(n - 2). In order to solve for n in terms of S, we must isolate the n on one side.

S = 180(n - 2) ⇒ divide both sides by 180

S ÷ 180 = n - 2 ⇒ add 2 to both sides

2 + (S ÷ 180) = n

Therefore, n = 2 + (S ÷ 180)

Repeat the procedure of dividing, multiplying, subtracting, and bringing the next digit down until you have solved the division problem's quotient.

For example, consider the long-division problem:

4567 ÷  123

In this problem, we need to repeat the process of dividing, multiplying, subtracting, and bringing the next digit down three times, since the divisor has three digits.

However, in some cases, you may need to repeat the process more or fewer times than the number of digits in the divisor.

For example, if the dividend has fewer digits than the divisor, you may need to repeat the process fewer times. Or, if you need to carry a digit over to the next step, you may need to repeat the process more times.

Thus, in general, you should repeat the process of dividing, multiplying, subtracting, and bringing the next digit down until you have found the quotient of the division problem. This will typically be the number of digits in the divisor, minus one, but it may be more or fewer in some cases.

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Answer:

Step-by-step explanation:

The equation of line in slope intercept form is, [tex]y=\frac{3}{4}x+9[/tex].

What is slope point form of the line?

A line's point slope form equation is y - y1 = m. (x – x1). Consequently,

y - 0 = m(x = 0), or y = mx, is the equation of a line passing through the origin with a slope of m.

Let the line passes through the point (-4, 6) and having slope 3/4.

We know that the equation of line passes through the point [tex](x_1, y_1)[/tex] and having slope m is,

[tex]y-y_1=m(x-x_1)[/tex]

Here, [tex](x_1, y_1) = (-4, 6)[/tex]  and slope [tex]m = \frac{3}{4}[/tex]

Plug the values in the above equation.

⇒

[tex]y-6=\frac{3}{4}(x-(-4))\\ y - 6 = \frac{3}{4}(x + 4)\\ y-6=\frac{3}{4} x + 3\\y = \frac{3}{4}x+3+6\\ y=\frac{3}{4}x+9[/tex]

Hence, the equation of line in slope intercept form is, [tex]y=\frac{3}{4}x+9[/tex].

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Using AutoSum in Excel is one quick and simple method of adding numbers. Simply click in a blank cell immediately underneath a column of data. Click AutoSum > Sum next on the Formula tab.

Sum is the outcome of adding two or more numbers or quantities. There may be only two terms, or there may be one hundred, thousand, or a million. There are some summations that have an infinite number of terms.

here

Click AutoSum on the Home tab, choose a cell next to the numbers you wish to add, then hit Enter when you're finished.

When you choose AutoSum, Excel automatically creates a formula (using the SUM function) to add the values.

Simply click in a blank cell immediately underneath a column of data. Click AutoSum > Sum next on the Formula tab.

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We need to use the concept of Plotting of Graphs.

A.(2018,62) and (2022,74) are the given points.

B. Slope=3

C. Height of Deshaun will increase 3 inches per year.

You must comprehend how the coordinate plane is organized and be aware of what to do with those (x, y) coordinates in order to graph points on the coordinate plane. Simply follow these instructions to learn how to graph points on the coordinate plane. You will graph a point in (x, y) form when you are graphing it on the coordinate plane. What you must understand is as follows:

The second coordinate is on the y-axis, while the x-axis runs left to right.

The y-axis is oriented vertically.

Positive numbers increase or move right (depending on the axis). Negative numbers move to the left or down.

Slope of a line segment=y2-y1/x2-x1 where (x1,y1) and (x2,y2) are the given points.

2018, Deshaun was 62 inches tall.

2022, Deshaun was 74 inches tall.

So assume the age on X-axis and Height on Y-axis.

A)Label the points we could use to find slope from this information:

  (2018,62) and (2022,74).

B) We know that,

    Slope=y2-y1/x2-x1

    ⇒Slope=(74-62)/(2022-2018)

    ⇒ Slope=12/4

    Slope=3.

C) As per the given slope, the height of Deshaun will increase 3 inches per year.

We need to use the concept of Plotting of Graphs.

A.(2018,62) and (2022,74) are the given points.

B. Slope=3

C. Height of Deshaun will increase 3 inches per year.

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The commutative, associative, and distributive property are the properties of addition and multiplication.

Commutative Property - The Commutative Property is derived from the word commute, which means to move. It states that the product or sum of two numbers do not change even if the places of numbers are changed or if the numbers move from one place to another. The commutative property of addition states that a + b = b + a and the commutative property of multiplication states that a × b = b ×  a.

Associative Property - It is related with the grouping of numbers. It states that the product or sum of three numbers do not change even if the grouping of numbers are changed. The associative property of addition states that (a + b) + c = a + (b + c) and the associative property of multiplication states that: (a × b) × c = a × (b ×  c).

Distributive Property - It is related with the distribution of numbers. It states that the product of three numbers do not change even if the distribution of numbers are changed. The distributive property of multiplication states that: a × (b + c) = (a × b) + (a ×  c).

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Ten bananas and ten pears will run you about £4.537.

Simultaneous equations are two equations with two unknown variables. From the given question;

If three bananas and two pears cost 95p and Five bananas and three pears cost £1.51, the equivalent equations will be:

3b + 2p = 0.95 * 3

5b + 3p = 1.51 * 2

Solve simultaneously;

6b + 6p = 2.85

10b + 6p = 3.02

Subtract

-4b = -0.17
b = 0.17/4

b = £0.0425

Since 3b + 2p = 0.95, hence

3(0.0425) + 2p = 0.95

0.1275 -0.95 = -2p

2p = 0.8225

p = 0.4112

The cost of  ten bananas and ten pears will be:

Cost = 10(0.0425) + 10(0.4112)
Cost  = 0.425 + 4.112
Cost = £4.537

Hence the cost of ten bananas and ten pears is approximately £4.537

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The coefficient of yA² for the three given expressions are 2, 10 and 5 respectively.

It is a mathematical statement that defines a mathematical concept as well as the system of a mathematical operation express in a symbolic form.

Mathematical expression is made up of at least two numerical terms along with a sign or at least two terms along with a sign. The expression having two or more numerical terms is called numerical expression.

As example 4 + 3 or 100÷4

The expression which has both variables and numbers is called algebraic expression.

As example X+6 or 25+5y

In an expression, the numbers which are accompanied by variables is called coefficient. As example, 24 + 3y, where 3 is the coefficient of y

but 24 is a constant number.

We are given three expressions, these are 2xA²y, 10xyA² and 5yA²

for the 1st expression, 2 is the coefficient of A²y

for the 2nd expression, 10 is the coefficient

for the last 5 is the coefficient.

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It is possible to compare the square roots of one hundred seventy and one hundred six eighths using the formulas sqrt(170) sqrt(106/8), and sqrt(170) = sqrt(106/8).

A number's square root is a value that yields the original number when multiplied by itself. The other way to square an integer is to find its square root. Squares and square roots are hence linked ideas.

Assuming that x is the square root of y, the equation would be written as x=y or as x2 = y. The radical symbol for the number's root is "" in this instance. The square of the positive number is represented by multiplying it by itself. The original number is obtained by taking the square root of the square of a positive number.

For comparing them, we should always keep in mind that if square or cube roots of two numbers ('a' and 'b') are to be compared, such that 'a' is greater than 'b', then a2 will be greater than b2 and a3 will be greater than b3 and so on, i.e., nth

The square root of 170 is approximately 13.038404810405015, while the square root of 106/8 is approximately 3.2403703492039303. Therefore, the square root of 170 is greater than the square root of 106/8. This can be written as:

sqrt(170) > sqrt(106/8)

Alternatively, you can compare the two numbers using the less than (<) or equal to (=) symbols as follows:

sqrt(170) < sqrt(106/8)

sqrt(170) = sqrt(106/8)

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The square root of an integer is a quantity that, when multiplied by itself, equals the original value. Finding an integer's square root is another technique to square it. Thus, squares and square roots are related concepts.

The equation would be expressed as x=y or as x2 = y if x were the square root of y. In this case, the radical symbol for the number's root is "". By multiplying the positive integer by itself, the square of the number is represented. By taking the square root of the square of a positive number, the original number can be found.

The square root of 170 is approximately 13.038404810405015, while the square root of 106/8 is approximately 3.2403703492039303. Therefore, the square root of 170 is greater than the square root of 106/8. This can be written as:

sqrt(170) > sqrt(106/8)

Alternatively, you can compare the two numbers using the less than (<) or equal to (=) symbols as follows:

sqrt(170) < sqrt(106/8)

sqrt(170) = sqrt(106/8)

When given two values in algebra, it is simple to find the third value. The x value is 5.

x Multiplicand Equals Product

X = product / multiplicand

* x = 10

Product

The x value is 5.

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Answer: C. 1200$

Step-by-step explanation:

$600 times twice is $1200

In the second year, the APR (Annual Percentage Rate) will increase quite substantially.

What is Introductory APR?

a temporary promotional interest rate that is lower than the card's standard APR, occasionally as low as 0% APR. Purchases, balance transfers, or both may be covered. After the introductory period has been over, your balance will be subject to the standard APR.

Any outstanding balances will begin to accrue interest once the promotional period is expired. Not just new charges, but also any amounts you charged or transferred to the credit card within the promotional APR period.

Your APR will transition from your promotional interest rate to a standard variable APR rate decided by your lender once it expires.

So, in this case, Trina's credit card has an introductory APR of 7.99-8.99%, but from the second year onwards the APR will increase.

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Following the conclusion of the promotional period, interest will start to be charged on any unpaid balances. Not only fresh charges, but also any sums you added to the credit card or transferred within the promotional APR period.

When the promotional period is up, your APR will change from the promotional interest rate to a standard variable APR rate chosen by your lender.

Thus, Trina's credit card in this instance has an introductory APR of 7.99–8.99%; but, in the second year, the APR will rise.

(x-3.5), or (x-7/2) are the factors of the polynomial, f(x) = 6x⁴ – 21x³ – 4x² + 24x – 35

Given,

The polynomial, f(x) = 6x⁴ – 21x³ – 4x² + 24x – 35

We have to find the factors of the polynomial;

Here,

This polygon's 35-factor factors, plus or minus 1, plus or minus 5, plus or minus 7, are potential zeros. Determine whether or not there is a non-zero remainder in each example using synthetic division. If none of these solutions work, try rational divisors derived from 35 and 6 such as 5/6, 7/6, 1/6 and so on.

Now,

f(x) = 6x⁴ – 21x³ – 4x² + 24x – 35

If used correctly, this method will eventually reveal one or two zeros of this poly. Of course, it would be lot simpler if you looked at the potential solutions provided to you for this issue.

I plotted this function and discovered that it crosses the x-axis at 7/2. Another root exists.

To determine whether or whether 7/2 is a root, use synth. div.

Then,

        ___________________________

7/2   /   6    -21    -4    24    -35

                  21      0   -14     35          

       ----------- ------------------------------

          6        0     -4    10       0

Since there is no remainder, the polynomial has a root of 7/2 (or 3.5). Therefore, a factor is (x-3.5) or (x-7/2).

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The coefficient of x² in (3x² – 5)( 4 +4x²) is -8.

A number multiplied by a variable is referred to as the coefficient of a number. For instance, the x2 coefficient in the formula 6x2+99 is 6.

First, let's multiply the terms in the given expression by opening the brackets.

⇒   (3x² – 5)( 4 +4x²)

Now, further multiplying we would get,

⇒   3x²(4 + 4x²) -5(4 + 4x²)

⇒    12x² +12x⁴-20-20x²

⇒  12x⁴ -8x²-20

As we can see that the term that is in multiplication with x² is -8. Similarly, the term that is coefficient of  x⁴ is +12.

Hence, the coefficient of x² in the equation  is -8.

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