[tex]a_1=2;\ a_2=-6;\ a_3=18;\ a_4=-54;\ a_5=162;\ ...[/tex]
-----------------------------------------------------
A recursive rule for a geometric sequence:
[tex]a_1\\\\a_n=r\cdot a_{n-1}[/tex]
[tex]r=\dfrac{a_{n+1}}{a_n}\to r=\dfrac{a_2}{a_1}=\dfrac{a_3}{a_2}=\dfrac{a_4}{a_3}=...\\\\r=\dfrac{-6}{2}=-3[/tex]
Therefore [tex]\boxed{a_1=2;\qquad a_n=-3a_{n-1}}[/tex]
-----------------------------------------------------
The exciplit rule:
[tex]a_n=a_1r^{n-1}[/tex]
Substitute:
[tex]a_n=2(-3)^{n-1}=2(3)^n(3)^{-1}=2(3)^n\left(\dfrac{1}{3}\right)\\\\\boxed{a_n=\dfrac{2}{3}\left(-3)^n}[/tex]
A student raises her average grade from a 75 to a 90. What is the percent of the increase in the students average grade
A recipe calls for 1 1/2 cups of sugar. Marley only has 1/4 cup to measure. How many times will Marley ave to fill up measuring cup?
What is the slope of the line that passes through the points (8,8) and (20,−2)? Write your answer in simplest form.
Answer:
[tex]\displaystyle m=\frac{-5}{6}[/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]\displaystyle m=\frac{y_2-y_1}{x_2-x_1}[/tex]Step-by-step explanation:
Step 1: Define
Point (8, 8)
Point (20, -2)
Step 2: Find slope m
Simply plug in the 2 coordinates into the slope formula to find slope m
Substitute [SF]: [tex]\displaystyle m=\frac{-2-8}{20-8}[/tex][Fraction] Subtract: [tex]\displaystyle m=\frac{-10}{12}[/tex][Fraction] Simplify: [tex]\displaystyle m=\frac{-5}{6}[/tex]Kit bought a soft drink and a sandwich for 9.00 what was the price of each ifnthe sandwich cost 3.5 times as much as the soft drink
Answer:
Sandwich = 6.75
Soft drink = 2.25
Step-by-step explanation:
9.00/4 = 2.25
2.25*3=6.75
Which of the vectors in the graph below is the negative of the vector v?
Option: C is the correct answer.
C. Vector d
Step-by-step explanation:Vector--
We know that the vector is used to represent a quantity which have both magnitude as well as direction.
The vector is represented in a diagram with the help of a line segment with an arrow.
We know that the negative of a vector reverses the direction of the original vector i.e. it directs to the opposite direction of the original vector.
i.e. both have the same starting point but the direction changes.
Here by looking at the figure we observe that the negative of the vector v is the vector d.
Graph y=3x^2-12x+13. What is the minimum value of the function?
Answer:
(2,1)
Step-by-step explanation:
You are only asking that the graph be drawn and that the minimum be stated. You are not asking how the minimum was obtained (by completing the square).
The graph (using Desmos) shows that the min is at (2,1)
Graphing is a mighty powerful tool, wouldn't you say? It tells you the answer before you have to do one bit of algebra.
Find the midpoint M of the line segment joining the points S = (−8, −1) and
T = (−4, −7)
Answer:
The midpoint is (-6,-4)
Step-by-step explanation:
The formula for the midpoint is
midpoint = (x1+x2)/2, (y1+y2)/2
Substituting what we know
= (-8+-4)/2, (-1+-7)/2
= -12/2, -8/2
= -6, -4
Final answer:
The midpoint M of the line segment joining points S = (-8, -1) and T = (-4, -7) is calculated using the midpoint formula and is found to be M = (-6, -4).
Explanation:
The midpoint M of the line segment joining the points S and T can be found using the midpoint formula. To find the midpoint M between the points S = (-8, -1) and T = (-4, -7), you need to take the average of the x-coordinates and the y-coordinates separately.
The midpoint M's x-coordinate is the average of the x-coordinates of S and T: ((-8 + (-4)) / 2 = -6. Similarly, the midpoint M's y-coordinate is the average of the y-coordinates of S and T: ((-1 + (-7)) / 2 = -4.
Therefore, the midpoint M is located at (-6, -4).
Using the information in the diagram, you can prove that segment WY ≅ segment ZX. Which reason would not appear in the proof?
AAS Congruence Theorem
Right Angles Congruence Theorem
SAS Congruence Postulate
Alternate Interior Angles Theorem
Answer: SAS congruence (choice C)
==============================================
Explanation:
Let's go through the various answer choices. I'll say whether or not they are used
A) This is used because angle WXZ = angle WYZ and angle WZX = angle ZWY are the two pairs of angles. The pair of sides are WZ = WZ which overlap (aka shared) between the two triangles. Note how the sides are not between the angled mentioned. Eg: for the triangle on the left, WZ is not between angle WXZ and angle WZX. So we can cross choice A off the list. Note: AAS is slightly different from ASA. Make sure not to confuse the two.
B) We use this to prove that the right angles (WXZ and WYZ) are congruent to each other. Both are 90 degrees. This will then feed in to choice A above, to help use AAS. We can cross choice B off the list.
C) We do not use SAS because we use AAS instead. We only have info about one pair of sides (WZ = WZ). We do not have info about another pair of sides. Therefore, this is the answer.
D) We use this fact to help set up that angle WZX = angle ZWY. The segments WY and XZ are parallel. I like to think of them as train tracks. On the inside of the train tracks are the angles WZX and ZWY, and they are on opposite sides of the transversal segment WZ, so this is why the pair of angles are alternate interior angles. They are congruent as long as WY is parallel to XZ. Like with choice B, this helps feed into choice A. We can cross choice D off the list.
Among the listed reasons, the Alternate Interior Angles Theorem would typically not appear in a proof proving that segment WY is congruent to segment ZX, unless the diagram involved parallel lines cut by a transversal. Other postulates might be used if angle and side measurements are congruent.
Explanation:In proving that segment WY ≅ segment ZX, some postulates or theorems may not appear in the proof depending on the given diagram. The Angel-Angle-Side (AAS) Congruence Theorem, the Side-Angle-Side (SAS) Congruence Postulate, and the Right Angles Congruence Theorem could all potentially appear in the proof if relevant angle and side measurements are given in the diagram.
However, the Alternate Interior Angles Theorem would typically not appear in this proof. This theorem is related to the indication that two lines are parallel when they are cut by a transversal and the alternate interior angles are congruent. This theorem does not directly involve proving the congruity of sides (or segments) as would be necessary for proving segment WY ≅ segment ZX. Unless the diagram specifically involved parallel lines cut by a transversal, this theorem would not be used.
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It costs Guido $0.20 to send a text message from his cell phone. He has already spent $4 in text messages this month. If he has a total of $10 that he can spend this month on text messages, write and solve an inequality that will give the greatest number of text messages that he can send. Interpret the solution.
The greatest number he can spend on is $5 since he has $10 he will have $5 reamining.
Dora cut a piece of string that is 33 centimeters long.What is the length of the piece of the string in meters
Answer:
0.33
Step-by-step explanation:
Factor [tex]x^3-4x^2-3x+18=0[/tex] given that 3 is a zero.
A) [tex](x+2)(x-3)^2=0[/tex]
B) [tex](x-2)(x-3)^2=0[/tex]
C) [tex](x-2)(x-3)(x+3)=0[/tex]
D) [tex](x+2)(x-\sqrt{3})(x+\sqrt{3})=0[/tex]
Answer:
A) (x +2)(x -3)² = 0
Step-by-step explanation:
Synthetic division by (x-3) gives the quadratic x² -x -6, which has factors (x -3) and (x +2). This is confirmed by another round of synthetic division.
The resulting factorization is ...
... (x +2)(x -3)² = 0
_____
A graph confirms a double zero at x=3 and one at x=-2.
Will give you brainliest!!
Answer:
Final answer is 7.
Step-by-step explanation:
Given that f(x) =7x+8.
Now we need to find the derivative of f(x) at x=5
So let's find derivative first.
[tex]f(x)=7x+8[/tex]
[tex]f'(x)=\frac{d}{dx}(7x+8)[/tex]
[tex]f'(x)=\frac{d}{dx}(7x)+\frac{d}{dx}(8)[/tex] {Using formula [f+g]'=f'+g'] }
[tex]f'(x)=7\frac{d}{dx}(x)+\frac{d}{dx}(8)[/tex] {Using formula f'(ax)=a f'(x) }
[tex]f'(x)=7(1)+0[/tex] {derivative of constant term is 0 }
[tex]f'(x)=7[/tex]
Now plug the given value of x=5
[tex]f'(5)=7[/tex]
Hence final answer is 7.
Simplify.
(2m^3n^2)5
[tex]\text{Use}\\\\(ab)^n=a^nb^n\\\\(a^n)^m=a^{n\cdot m}\\------------------\\\left(2m^3n^2\right)^5=2^5(m^3)^5(n^2)^5=32m^{3\cdot5}n^{2\cdot5}=32m^{15}n^{10}[/tex]
Solve the system of linear equations by substitution.
2x=y−10
x+7=y
Answer:
-3,4
Step-by-step explanation:
The solution to the system of linear equations 2x = y - 10 and x + 7 = y by substitution is (x, y) = (-3, 4).
Explanation:To solve the system of linear equations 2x = y - 10 and x + 7 = y by substitution, follow these steps:
First, solve one of the equations for one variable. From the second equation, we get x = y - 7.Next, substitute this expression for x into the first equation: 2(y - 7) = y - 10.Simplify and solve for y: 2y - 14 = y - 10, which simplifies to y = 4.Finally, substitute y back into one of the original equations to find x: x = 4 - 7, which simplifies to x = -3.The solution to the system of equations is (x, y) = (-3, 4).
Solve for xx. Write the smaller solution first, and the larger solution second. X^2 + 12x + 27 = 0x 2 +12x+27=0
Answer:
x = -9, -3
Step-by-step explanation:
x^2 + 12x + 27 = 0
We need 2 numbers whose product is + 27 and whose sum is + 12. These are 3 and 9. So the factors are:-
(x + 3)(x + 9) and this = 0
So (x + 3) = 0 or x + 9 = 0
x = -3 , -9
Answer:
-5 and 7
Step-by-step explanation:
we need to find numbers a and b such that a+b=-2 and ab=-35.
a=5 and b=-7 satisfy both conditions, so our equation can be re-written:
(x + 5)(x-7) = 0
According to the zero-product property, we know that
x+5=0 or x-7=0, which means
x=-5x or x=7
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Simplify f+g / f-g when f(x)= x-4 / x+9 and g(x)= x-9 / x+4
[tex]f(x)=\dfrac{x-4}{x+9};\ g(x)=\dfrac{x-9}{x+4}\\\\f(x)+g(x)=\dfrac{x-4}{x+9}+\dfrac{x-9}{x+4}=\dfrac{(x-4)(x+4)+(x-9)(x+9)}{(x+9)(x+4)}\\\\\text{use}\ a^2-b^2=(a-b)(a+b)\\\\=\dfrac{x^2-4^2+x^2-9^2}{(x+9)(x+4)}=\dfrac{2x^2-16-81}{(x+9)(x+4)}=\dfrac{2x^2-97}{(x+9)(x+4)}\\\\f(x)-g(x)=\dfrac{x-4}{x+9}-\dfrac{x-9}{x+4}=\dfrac{(x-4)(x+4)-(x-9)(x+9)}{(x+9)(x+4)}\\\\\text{use}\ a^2-b^2=(a-b)(a+b)\\\\=\dfrac{x^2-4^2-(x^2-9^2)}{(x+9)(x+4)}=\dfrac{x^2-16-x^2+81}{(x+9)(x+4)}=\dfrac{65}{(x+9)(x+4)}[/tex]
[tex]\dfrac{f+g}{f-g}=(f+g):(f-g)=\dfrac{2x^2-97}{(x+9)(x+4)}:\dfrac{65}{(x+9)(x+4)}\\\\=\dfrac{2x^2-97}{(x+9)(x+4)}\cdot\dfrac{(x+9)(x+4)}{65}\\\\Answer:\ \boxed{\dfrac{f+g}{f-g}=\dfrac{2x^2-97}{65}}[/tex]
The value of f+g / f-g = (2x² - 97) / 65
What is Function?In mathematics, a function is represented as a rule that produces a distinct result for each input x. In mathematics, a function is indicated by a mapping or transformation. Typically, these functions are identified by letters like f, g, and h. The collection of all the values that the function may input while it is defined is known as the domain. The entire set of values that the function's output can produce is referred to as the range. The set of values that could be a function's outputs is known as the co-domain.
Given:
f(x)= x-4 / x+9 and g(x)= x-9 / x+4
So, f+g / f-g
f(x)+ g(x)
= x-4 / x+9 + x-9 / x+4
= (x² - 4²) + (x² - 9²) / (x+4)(x+9)
= (2x² - 97) / (x+4)(x+9)
f(x)-g(x)
= x-4 / x+9 - x-9 / x+4
= (x² - 4²) - (x² - 9²) / (x+4)(x+9)
= 65 / (x+4)(x+9)
Then, f+g / f-g
= (2x² - 97) / (x+4)(x+9) x (x+4)(x+9)/ 65
= (2x² - 97) / 65
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Can someone help with these? 15 PTS
Answer: D/3f^2 = e
Step-by-step explanation:
D = 3ef^2
Divide by 3f^2 on both sides.
D/3f^2=e, the first option.
What does the fundamental theorem of algebra state about the equation 2x^2−4x+16=0 ?
Answer:
option B
Step-by-step explanation:
[tex]2x^2-4x+16=0[/tex]
We need to solve this equation using quadratic formula
[tex]x= \frac{-b+-\sqrt{b^2-4ac}}{2a}[/tex]
a=2, b= -4, c=16
Plug in the values in the formula
[tex]x= \frac{-(-4)+-\sqrt{(-4)^2-4(2)(16}}{2(2)}[/tex]
[tex]x= \frac{4+-\sqrt{16-128}}{4}[/tex]
[tex]x= \frac{4+-\sqrt{-112}}{4}[/tex]
Simplify the square root. the value of square root (-1) = 'i'
[tex]x= \frac{4+-4i\sqrt{7}}{4}[/tex]
Now we divide by 4
[tex]x= 1+-i\sqrt{7}[/tex]
So there are two complex roots. since the degree of polynomial is 2
Answer:
B
Step-by-step explanation:
I took the test :)
The length of a rectangle is six times its width. If the area of the rectangle is 486 in 2 , find its perimeter.
w - width
6w - length
486 in² - area of the rectangle
(w)(6w) = 6w² - area of the rectangle
The equation:
6w² = 486 divide both sides by 6
w² = 81 → w = √81
w = 9 in
l = 6w → l = (6)(9) → l = 54 in
The perimeter: P = 2l + 2w
P = 2(54) + 2(9) = 108 + 18 = 126 in
Answer: The perimeter is 126inFinal answer:
The rectangle with an area of 486 square inches has a width of 9 inches and a length of 54 inches, leading to a perimeter of 126 inches.
Explanation:
To find the perimeter of a rectangle when given the area and the relationship between length and width, first, set up an equation where length (L) is equal to six times the width (W), so L = 6W. The area of a rectangle is found by multiplying the length by width, so we have the equation Area = L × W. With the given area of 486 square inches, we can substitute L with 6W to get 486 = 6W × W.
Solving this equation, W2 = 486/6 leads to W2 = 81. Taking the square root, we find that W = 9 inches. Since the length is six times the width, L = 6 × 9 = 54 inches.
The perimeter (P) of a rectangle is the sum of twice the length plus twice the width, so the equation P = 2L + 2W can be used. Substituting the values we found, P = 2(54) + 2(9) leads to P = 108 + 18, so the perimeter equals 126 inches.
Jason is selling video games. To earn his monthly bonus, he must sell a minimum of 5 games. He has 30 he can sell. The video games cost $20 each. The function f(x) = 20x can be used to represent this situation. What is the practical range of the function?
All whole numbers from 5 to 30, inclusive.
All whole numbers from 100 to 600, inclusive.
All real numbers.
All multiples of 20 between 100 and 600, inclusive.
Answer:
Correct choice is B
Step-by-step explanation:
The function [tex]f(x)=20x[/tex] represents the situation, where x is the number of sold video games and f(x) is the total cost of sold games.
Jason must sell a minimum of 5 games, this means that [tex]x\ge 5.[/tex] He has 30 video games he can sell, then [tex]x\le 30.[/tex] Thus, the domain of the function is [tex]5\le x\le 30.[/tex]
The range of the function f(x) is
[tex]20\cdot 5\le f(x)\le 20\cdot 30,\\ \\100\le f(x)\le 600.[/tex]
Congruency statements
ΔABC≅ΔDEF
List all pairings of congruent angles & sides for the above congruence statement.
ΔABC ≅ ΔDEF
I find it easier to see the congruencies if I write them underneath each other:
ABCDEF∠A ≅ ∠D [tex]\overline{AB}[/tex] ≅ [tex]\overline{DE}[/tex]
∠B ≅ ∠E [tex]\overline{BC}[/tex] ≅ [tex]\overline{EF}[/tex]
∠C ≅ ∠F [tex]\overline{AC}[/tex] ≅ [tex]\overline{DF}[/tex]
What is the expected number of pets for a student in his class?
Answer:
A) about 1 pet
Step-by-step explanation:
Total the products of number of pets and their probability.
... 0×0.4 +1×0.3 +2×0.25 +3×0.05
... = 0 +0.3 +0.5 +0.15 = 0.95 . . . . . about 1
___
We assume that "3 or more" will be closer to 3 than to 15, a number that would raise the expected value to more than 1.5.
Ther are 330 walnuts in 6 bags if each bag has the same number of walnuts how many walnuts are in 2 bags
Answer:
110 Walnuts
Step-by-step explanation:
Firstly, there are 6 bags and 330 walnuts. You divide 330 by 6 and you get 55. 330/6. Now, you multiply 55 by 2 and you get your answer. So basically in each bag there are 55 walnuts and if you multiply that by 6 you get 330 again. Hope this helped!
Answer:
110 walnuts
Step-by-step explanation:
330÷6 is 55
55•2 is 110
Joe has 12 cups of soup in a pot. If he puts 1/4 cup of soup into each bowl, how many bowls will be used?
Answer:
48
Step-by-step explanation:
you need 4 one-fourth cups to make one cup. multiply 12 by 4 and you get 48
By dividing the total amount of soup (12 cups) by the amount of soup per bowl (1/4 cup), we find that Joe can fill 48 bowls.
The student poses a question related to division and unit conversion. To address the student's question, we should calculate how many bowls Joe can fill by dividing the total amount of soup by the amount used for each bowl. Joe has 12 cups of soup and uses 1/4 cup of soup for each bowl. The calculation would be:
Number of bowls = Total cups of soup / Cups of soup per bowl
Number of bowls = 12 cups / 1/4 cup per bowl
Number of bowls = 12 / 0.25
Number of bowls = 48
Therefore, Joe will use 48 bowls to serve the 12 cups of soup, with 1/4 cup of soup in each bowl.
Consider that the system has infinitely many solutions. Which choice is the coefficient of y in the second equation? 5x + 10y = 15 x + __ y = 3
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The formula for the circumference of a circle is = 2. Use 3.14 for . Solve the formula for r. What is the radius of a circle with a circumference of 87 ? Round to the nearest tenth.
a. = 2; 13.9 cm
b. =2 π ; 27.7 cm
c. = 2 ; 546.4 cm
d. = − 2 ; 80.7 cm
The formula for the area of a triangle is = ℎ /2 Solve the formula for h. What is the height of a triangle that has an area of 25 2 and a base with a length of 10 ?
a. ℎ = 2 ; 1.25 mi
b. ℎ = 2 ; 2.5 mi
c. ℎ =2/; 0.2 mi
d. ℎ = 2/; 5 mi
Answer:
C / (2pi) = r; r= 13.9 cm
2A/b = h ; h= 5
Step-by-step explanation:
The formula for the circumference of a circle is C = 2* pi * r.
Solving for r, we need to divide by 2 * pi on each side
C/(2* pi) = 2*pi*r/ (2* pi)
C / (2pi) = r
If the circumference is 87, we can find r by substituting into the equation.
87 / (2 * 3.14) = r
87/ 6.28 = r
13.85350318=r
Rounding to the nearest tenth
r= 13.9 cm
The formula for the area of a triangle is A = bℎ /2
To solve the formula for h, we need to multiply each side by 2
2*A = bh
Then divide each side by b
2A/b = bh/b
2A/b = h
If the area of a triangle is 25 and the base of 10 , we can substitute in to find the height.
2* 25 /10 = h
50/10 = h
5 =h
An organization of berry farmers releases a study reporting that people who eat blueberries every day have a lower cholesterol level. Which statement describes the best conclusion to draw from the study?
Answer:
There may be a correlation between eating blueberries and a lower cholesterol level.
Step-by-step explanation:
cause correlation does not cause causation.
What is the end behavior of the graph?
Answer:
b
Step-by-step explanation:
Looking at the graph, as x gets more and more negative, y gets larger and larger.
x→-∞ y →∞
As x approaches negative infinity, y approaches infinity
As x gets larger and larger, y gets more and more negative
x→∞, y→-∞
As x approaches infinity, y approaches negative infinity
The correct answer is B) "As x approaches infinity, f(x) approaches negative infinity; as x approaches negative infinity, f(x) approaches infinity."
"As x approaches infinity, function f(x) approaches negative infinity": When x becomes very large (positive infinity), the term (3/17)x dominates the function. Since the coefficient 3/17 is positive, as x increases towards infinity, (3/17)x also increases without bound, causing f(x) to approach negative infinity.
"As x approaches negative infinity, function f(x) approaches infinity": Similarly, as x becomes very large in the negative direction (negative infinity), the term (3/17)x becomes very negative. Again, since the coefficient 3/17 is positive, f(x) increases without bound (but in the positive direction) as x approaches negative infinity.
In summary, the end behavior of the graph is such that as x goes to positive infinity, f(x) goes to negative infinity, and as x goes to negative infinity, f(x) goes to positive infinity, which corresponds to option B.
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Patti green and her husband are purchasing a condominium for $123,000. They have $15,000 for a down payment. What is the amount of their mortgage loan?
Answer:
The amount of their mortgage loan is $108000
Step-by-step explanation:
we are given
total purchasing amount =$123000
down payment amount = $15000
we know that
Mortgage loan amount = (total purchasing amount)-(down payment amount)
now, we can plug value
Mortgage loan amount = $123000-$15000
Mortgage loan amount =$108000
Patti Green and her husband need a mortgage loan of $108,000 for their condominium purchase.
The student has asked for the amount of the mortgage loan that Patti Green and her husband would need for purchasing a condominium. To find this, we would subtract their down payment from the total cost of the condominium. Since the condominium costs $123,000 and they have a $15,000 down payment, we do the following calculation:
Total cost of the condominium = $123,000
Down payment = $15,000
Mortgage loan needed = Total cost of the condominium - Down payment
Mortgage loan needed = $123,000 - $15,000
Mortgage loan needed = $108,000
Therefore, the amount of the mortgage loan that Patti Green and her husband would need is $108,000.
What is the value of x in the equation 2^2x = 2^3?
A.) x = 1.5
B.) x = 3
C.) x = 64
D.) x = 0.28
Answer:
The value of x is A) 1.5
Step-by-step explanation:
In order to find this, we can first note that if the base is the same, their exponents also must be the same. Since both have a base of 2, then we can eliminate the base and simply compare their exponents.
2x = 3
Now we can solve by simply dividing both by 2.
x = 1.5