Using the principle of mathematical induction, the [tex]n^{th }[/tex] term of the given quadratic sequence is found to be 3n² - n + 1.
What exactly is mathematical induction?Mathematical induction is a technique for proving the truth of a statement, theorem, or formula for each and every natural number n. This is generalized as the 'Principle of Mathematical Induction,' which we would use to prove any mathematical statement.Given sequence: 3, 11, 25, 45,...
The first term of the sequence is 3, the second term is 11, the third term is 25, and the fourth term is 45.
The difference between the first and second terms can be calculated as follows:
11-3 = 8
The difference between the second and third terms can be calculated as follows:
25-11 = 14
The difference between the third and fourth terms can be calculated as follows:
45-25 = 20
The sequence is expressed as follows:
3,3+8,11+11,25+20,...
The difference between consecutive terms expands by 6.
Use the principle of mathematical induction.
[tex]6(\frac{n(n+1)}{2})[/tex]
= 3n(n+1)
The sequence's nth term can be calculated as follows:
[tex]n^{th }[/tex] term = 3n(n+1) - 4n + 1
= 3n² - n + 1
Therefore, the [tex]n^{th }[/tex] term of the given quadratic sequence is found to be 3n² - n + 1 using the principle of mathematical induction.
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Complete the order pairs below so they satisfy the equation Y=-3x+2Complete the order pairs (0,__) (7,__) (__,17)
Help!!
The ordered pairs which satisfy the equation are (0,2) , (7,19) (-5,17) .
Finding the values of the variables that result in equality is the first step in solving a variable equation.
The values of the unknown variables that fulfil the equality are the equation's solutions, also known as the variables for which the equation must be solved. The two forms of equations are identity equations and conditional equations. All feasible values of the variables share the same identity. Only specific instances where the values of the variables coincide can result in the truth of a conditional equation. Two phrases are combined into one by using the equals sign ("="). The "left hand side" and "right hand side" of the equation refer to the expressions on each side of the equals sign. It's typical.At x = 0 the value of y from the equation is :
y = 2
At x = 7 , y =-19
at y=17 , x= -5
Therefore the ordered pairs are :(0,2) , (7,19) (-5,17) .
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Please explain how you got the question because i have no idea how to plug in the numbers
Given sequence:
[tex]\frac{1}{2},\text{ }1,\text{ }\frac{3}{2},2[/tex]To find the 109th term, we need to first determine whether the sequence follows an arithmetic progression or a geometric progression.
Check
The sequence is arithmetic if the successive terms of the sequence are formed by adding or subtracting a value.
The sequence is geometric if the successive terms of the sequence are formed by multiplying or dividing by a value.
We have that:
first term = 1/2
second term = 1
third term = 3/2
Taking the difference of the second term from the first term:
[tex]\begin{gathered} =\text{ 1-}\frac{1}{2} \\ \text{ = }\frac{1}{2} \end{gathered}[/tex]Taking the difference of the third term from the second term:
[tex]undefined[/tex]What are the solutions to the equation 6x2 - x - 40 = 0? A. x = -8/3, x = -5/2B. x = -8/3, x = 5/2 С. x = -2, x = 3 D. x = -6, x = 2
Solve the following equation for all values of x:2 +3−28=9+27
A train can travel 225 kilometers in 3 hours. At this rate, how many kilometers can the train travel in 7 hours?
Answer: 1575
Step-by-step explanation: You can multiply 225x7 and you will get 1,575
Kai took five tests this semester. His scores are listed below. 92, 88, 90, 96, 84What is the mean absolute deviation of Kai's test scores?4.03.22.01.6
the mean can be calculate with this equation:
[tex]m=\frac{x_1+x_2+\cdots}{n}[/tex]So in our problem will be:
[tex]\begin{gathered} m=\frac{92+88+90+96+84}{5} \\ m=90 \end{gathered}[/tex]3. Find the volume of the
cylinder. Round to the nearest
tenth. Diameter = 15. Hight = 11
Answer:
518.4
Step-by-step explanation:
the volume of a cylinder is area of base x height
the radius is half of the diameter = 7.5
area of base = 2πr = 2xπx7.5 = 47.124
volume = 47.124 x 11 = 518.4
The radius of a circular pond is increasing at a constant rate, which can be modeled by the function r(t) = 5t, where tis time in months. Thearea of the pond is modeled by the function A(1) = 1+?. The area of the pond with respect to time can be modeled by the compositionA(r(t).Which function represents the area with respect to time?OAA(r(t) = 5x52OBA(r(t) = 25m€2O cA(r(E) = 10x92O D.A(-(t)) = 51t?
Explanation
We are asked to find the function that represents the area with respect to time A(r(t))
Given that
[tex]\begin{gathered} r(t)=5t \\ \\ A(r)=\pi r^2 \end{gathered}[/tex]what we will simply do will be to put in the value of r(t) into the equation of A(r)
Thus
[tex]\begin{gathered} A(r)=[\pi\times(5t)^2] \\ \\ A(r)=\pi\times25t^2 \\ \\ A(r)=25\pi t^2 \end{gathered}[/tex]Thus, the answer is option B
there are 4 swimmers in a race. there will be 1st, 2nd and 3rd place prize awarded. In how many different ways can the prizes be awarded?
From the problem we will have that:
[tex]\begin{gathered} _4P_3=\frac{4!}{(4-3)!}\Rightarrow_4P_3=\frac{24}{(1)} \\ \\ \Rightarrow_4P_3=24 \end{gathered}[/tex]There will be 24 possible ways to award the 3 prizes between 4 people.
.shvsguvsguvsguvsguss
Answer:
what is that ? What do you mean
Answer:
m
Step-by-step explanation:
What are the solutions to the equation 3|x + 5| - 2 = 13 ?
To solve the absolute value equation;
[tex]3|x+5|-2=13[/tex]Note that the left side of the equation is an absolute value. The first step is to remove the absolute value sign, and then the next step is to solve while using the positive and negative value of the number on the right side of the equation.
This is shown below;
[tex]\begin{gathered} 3|x+5|-2=13 \\ \text{remove the absolute value sign;} \\ We\text{ now have;} \\ 3(x+5)-2=13 \\ 3(x+5)=13+2 \\ 3x+15=15 \\ \text{Subtract 15 from both sides;} \\ 3x+15-15=15-15 \\ 3x=0 \\ \text{Divide both sides by 3} \\ \frac{3x}{3}=\frac{0}{3} \\ x=0 \end{gathered}[/tex]Let us now solve for the equation when the right side is -13.
[tex]\begin{gathered} 3(x+5)-2=-13 \\ 3(x+5)=-13+2 \\ 3x+15=-11 \\ \text{Subtract 15 from both sides;} \\ 3x+15-15=-11-15 \\ 3x=-26 \\ \text{Divide both sides by 3;} \\ \frac{3x}{3}=-\frac{26}{3} \end{gathered}[/tex]ANSWER:
[tex]\begin{gathered} x=0, \\ OR \\ x=-\frac{26}{3} \end{gathered}[/tex]The last option is the correct answer
Which best describes the relationship between the lines with equations 4x−8y=9 and 8x−7y=9?A. parallelB. same lineC. neither perpendicular nor parallelD. perpendicular
The given equations are,
4x-8y=9 and 8x-7y=9
The slopes can be determined as,
[tex]\begin{gathered} m_1=\frac{-a}{b}=\frac{-(4)}{-8}=\frac{1}{2} \\ m_2=\frac{-a}{b}=\frac{-(8)}{-7}=\frac{8}{7} \end{gathered}[/tex]As, the slope are not equal nor their product is -1.
Thus, they are neither parallel nor perpendicular.
Thus, option (C) is correct.
A particular fruit's weights are normally distributed, with a mean of 406 grams and a standard deviation of 27 grams.The heaviest 14% of fruits weigh more than how many grams?Give your answer to the nearest gram.
Given
mean of 406 grams and a standard deviation of 27 grams.
Find
The heaviest 14% of fruits weigh more than how many grams?
Explanation
given
mean = 406 gms
standard deviation = 27 gms
using standard normal table ,
[tex]\begin{gathered} P(Z>z)=14\% \\ 1-P(Zso , [tex]\begin{gathered} x=z\times\sigma+\mu \\ x=1.08\times27+406 \\ x=435.16 \end{gathered}[/tex]Final Answer
Therefore , The heaviest 14% of fruits weigh more than 435.16 gms
An English teacher at a high school can teach six classes. There are 43 students enrolled in English I, 51 in English II, and 53 in English III. Use the Webster method to determine the apportionment of students to the courses to determine the number of sections needed per course.
Given:
Heigh school teach = 6 classes
Enrolled student
English 1 = 43
English 2 =51
English 3 =53
Apportionment student is:
[tex]\begin{gathered} =\frac{\text{English}1+\text{English}2+\text{English}3}{6} \\ =\frac{43+51+53}{6} \\ =\frac{147}{6} \\ =24.5 \end{gathered}[/tex]Solve for the unknown: K/2+3=5
Solve for K:
[tex]\frac{K}{2}+3=5[/tex]Subtracting 3 to each member of the equation:
[tex]\frac{K}{2}+3-3=5-3[/tex]Operating:
[tex]\frac{K}{2}=2[/tex]Now we multiply by 2:
[tex]2\cdot\frac{K}{2}=2\cdot2[/tex]Simplifying and operating:
[tex]K=4[/tex]Answer: K = 4
help me pleaseeeeeeeeeeeeeeeeeeeeeee
thank you
Answer:
Domain: A, [1, 7]
Range: [-4, 2]
Step-by-step explanation:
The domain is the set of x-values and the range is the set of y-values.
this is geometry please help
Answer: 1st one is sometimes
2nd one is always
3rd one I think is always
4th one is always
Step-by-step explanation:
1st one the sum must equal 90 so either one could be greater
2nd one vertical angles always equal each other
3rd one if B is the midpoint it would be AB BC
4th one it will always equal 180
Liam is comparing the costs of phone plans at two different companies to determine which plan to buy. Company A charges $50 each month plus an additional $10 per gigabyte of data used. Company B charges $60 each month plus an additional $8 per gigabyte of data used. The following equation represents the cost of the phone plans for the two companies when they are equal, where g represents the amount of gigabytes of data used. 50+10g+60+8g Complete each of the statements below with the correct answer choice.
The cost of the phone plans will be the same for company A and company B when
5?10?15?18? gigabytes of data are used.
If company A changes the cost of data per gigabyte to
20?24?50?8? , then the two phone plans will never have the same cost.
If company B's initial monthly cost Increases by $10?Decreases by $10
and the cost of data per gigabyte Remains the same? Decreases by $2? Increases by $2, then the cost of the two phone plans will always be the same.
1. Using equations, the cost of the phone plans will be the same for company A and company B when A. 5 gigabytes of data are used.
2. If company A changes the cost of data per gigabyte to D. 8, then the two phone plans will never have the same cost.
3. If company B's initial monthly cost B. Decreases by $10 and the cost of data per gigabyte B. Decreases by $2, then the cost of the two phone plans will always be the same.
What is an equation?An equation is a mathematical statement that shows that two mathematical expressions are equal or equivalent in value.
Equations are represented using the equation sign (=) to establish the equality of two numerical values or mathematical expressions.
Company A Company B
Fixed charge $50 $60
Variable charge per gigabyte $10 $8
The equation to represent when the two plans are equal:
50 + 10g = 60 + 8g
10g - 8g = 60 - 50
2g = 10
g = 5
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In \triangle NOP,△NOP, \overline{PN}\cong \overline{OP} PN ≅ OP and \text{m}\angle P = 50^{\circ}.m∠P=50 ∘ . Find \text{m}\angle O.m∠O.
The value of the measure of ∠O will be 65°
What is mean by Triangle?
A triangle is a three sided polygon, which has three vertices and three angles which has the sum 180 degrees.
Given that;
In ΔNOP;
PN ≅ OP
So, ∠O = ∠N
And, Measure of ∠P = 50°
Now,
In the ΔNOP;
The sum of all the three angles are 180°.
The value of the measure of ∠O = x
We get;
∠O + ∠N + ∠P = 180°
Substitute ∠P = 50° and ∠O = ∠N = x we get;
x + x + 50 = 180
2x + 50 = 180
2x = 180 - 50
2x = 130
x = 65
Thus, The value of the measure of ∠O will be 65°.
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Answer please and thanks
A percentage of 100 · [p' - exp(- 4000 · k)] / p' of carbon-14 has decayed and lost in last 4000 years.
How to determine the remaining percentage of carbon-14 in ancient pollen
In this question we have the case of simple decay of a radioactive isotope (carbon-14) in time, which is represented by the following exponential formula:
p(t) = p' · exp(- k · t)
Where:
p' - Initial amount of pollen.p(t) - Current amount of pollen.k - Decay constant.t - Time, in years.If we know that t = 4000 years, then percentage of lost carbon-14 is:
p = 100 - 100 · p(t) / p'
p = 100 - [100 · exp(- 4000 · k)] / p'
p = 100 · {1 - [exp(- 4000 · k)] / p'}
p = 100 · [p' - exp(- 4000 · k)] / p'
An amount of 100 · [p' - exp(- 4000 · k)] / p' per cent has been decayed and lost.
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Use f(x) = 4x – 2 and g(x) = 5 – x^2 to evaluate the expression.(a)(f o g)(-2)(b)(g o f)(-2)
Given the functions:
[tex]\begin{gathered} f(x)=4x-2 \\ g(x)=5-x^2 \end{gathered}[/tex]We will find (f o g)(-2)
So, first, we will find the function (f o g)(x) as follows:
[tex]\begin{gathered} \mleft(f\circ g\mright)(x)=f\lbrack g(x)\rbrack=4(5-x^2)-2 \\ (f\circ g)(x)=20-4x^2-2 \\ (f\circ g)(x)=18-4x^2 \end{gathered}[/tex]Now, substitute with x = -2
so,
[tex](f\circ g)(-2)=18-4\cdot(-2)^2=18-4\cdot4=18-16=2[/tex]so, the answer to the part (a) (f o g)(-2) = 2
b) (g o f)(-2)
We will find (g o f)(x) as follows:
[tex](g\circ f)(x)=g\lbrack f(x)\rbrack=5-(4x-2)^2[/tex]Substitute with x = -2
So,
[tex]\begin{gathered} (g\circ f)(-2)=5-(4\cdot-2-2)^2 \\ =5-(-8-2)^2 \\ =5-(-10)^2 \\ =5-100 \\ =-95 \end{gathered}[/tex]So, the answer to the part (b): (g o f)(-2) = -95
First six non zero multiples of 21
Answer:
21, 42, 63, 84, 105, 126
Step-by-step explanation:
I'm doing topic about classifying numbers can you please help me with these problems?
According to this question we have to classify what type of number is 10.
The number 10 should be considered as a natural number, real number, rational number, integer number, whole number because of the following reason:
It is a rational and a integer number because because it can be expressed as the quotient of two integers 10/1
Also, it is a natural number because natural numbers are numbers which are used for counting and ordering, and 10 is an example of them.
Also, it is a real number. Real numbers are any number on the real line. So, 10 it is on the real line so, is considered a real number.
Also it is a whole number because it is a natural number.
what is the correct algebraic reprentation for a translation 5 units right and 4 units down?
We want to write the correct algebraic representation for a translation 5 units right and 4 units down.
Which means 5 units along the positive x axis and 4 units along the negative y axis.
So, we have;
[tex](x,y)\rightarrow(x+5,y-4)[/tex]Therefore, the correct algebraic
The average speed, s, in miles per hour that a student walks the 3 miles from home to school varies inversely as the number of hours, h that the student walks.The formula is given by s = 3/hAs the number of hours it takes the student to walk from home to school increases, what happens to the speed?
If the number of hours it takes the student to walk home increases, then the speed decreases
Let us see an example
if h = 1
and we increase h to be = 3
[tex]\begin{gathered} \text{when h =1} \\ \text{The sp}eed,\text{ s =}\frac{3}{1}=3milesperhour \end{gathered}[/tex][tex]\begin{gathered} \text{When h increases to 3} \\ s\text{ =}\frac{3}{3}=1mileper\text{ hour} \end{gathered}[/tex]3miles/hour is more than 1 mile/hour so hence the speed will decrease when the time increases
The answer is option A, the speed decreases
McMichael Inc. collects 25% of its sales on account in the month of the sale and 75% in the month following the sale. If sales on account are budgeted to be $523,000 fo September and $465,000 for October, what are the budgeted cash receipts from sales on account for October?
The budgeted cash receipts from sales on account for October is $479500.
Define sale.A sale is a deal in which two or more parties exchange goods or services for cash or other assets. A sale in the financial markets is an arrangement involving the cost of a security and its delivery for agreed-upon payment between a buyer and a seller. Sales in general business operations refer to any exchanges of money or value for the right to possess a good or get a service. Sales, in the context of accounting, refers to the income generated by a corporation via the selling of goods or services (net sales).
Given Data
McMichael Inc. collects 25% of its sales on account in the month of the sale and 75% in the month.
If sales on account are budgeted to be $523,000 to September and $465,000 for October.
We can respond as follows based on the data provided in the question:-
25% of sales from October plus 75% of sales from September equal the budgeted cash receipts from sales.
Sales revenue budgeted for in cash is
($523,000 (25%)) + ($465,000 (75%))
130750 + 348750
479500
The budgeted cash receipts from sales on account for October is $479500.
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Solve the formula v = pi r2 h solve for r
We can do the following steps to solve the given expression for r.
Step 1: We divide by πh from both sides.
[tex]\begin{gathered} v=\pi r^2h \\ \frac{v}{\pi h}=\frac{\pi r^2h}{\pi h} \\ \frac{v}{\pi h}=r^2 \end{gathered}[/tex]Step 2: We apply square root to both sides of the equation.
[tex]\begin{gathered} \sqrt{\frac{v}{\pi h}}=\sqrt{r^2} \\ \sqrt{\frac{v}{\pi h}}=r \end{gathered}[/tex]Answer[tex]r=\sqrt{\frac{v}{\pi h}}[/tex]Donovan walked a total of 5 3/8 miles on Sunday and Saturday if he walked 2 and 1/4 Mile on Saturday how many did he walk on Sunday
The total miles Donovan walked were 5 3/8, we know that he walked 2 1/4 saturday then:
[tex]\begin{gathered} 5\frac{3}{8}-2\frac{1}{4}=\frac{43}{8}-\frac{9}{4} \\ =\frac{43}{8}-\frac{18}{8} \\ =\frac{25}{8} \\ =3\frac{1}{8} \end{gathered}[/tex]He walked 3 1/8 miles on sunday.
Hii i forgot how to multiply fractions and I'm a bit rusty since the pandemic
Solution
The area of a rectangle is given as length(l) x width(w)
length = 2 1/4
width = 1 1/2
Hence substituting these values into the formula
[tex]\begin{gathered} \text{Area = 2}\frac{1}{4}\times1\frac{1}{2} \\ \text{Area = }\frac{(4\times2)+1}{4}\times\frac{(2\times1)+1}{2} \\ \text{Area = }\frac{9}{4}\times\frac{3}{2} \\ \text{Area = }\frac{27}{8}in^2 \\ \text{Area = 3}\frac{3}{8}in^2 \end{gathered}[/tex]Express x^2-3x+1 in the form (x-p)^2+q
By algebra properties, the equation x² - 3 · x + 1 in the form (x - p)² + q is equal to (x - 3 / 2)² - 5 / 4.
How to simplify a quadratic equation by completing the square
Completing the square is a method used to simplify quadratic equations of the form a · x² + b · x + c that are not perfect squares, where a part of it becomes into perfect square and is simplified. This can be found by using algebraic methods. The complete procedure is shown below:
x² - 3 · x + 1
(x² - 3 · x + 1) + 0
(x² - 3 · x + 1) + [9 / 4 + (- 9 / 4)]
(x² - 3 · x + 9 / 4) + [1 + (- 9 / 4)]
(x - 3 / 2)² - 5 / 4
The quadratic equation is equivalent to (x - 3 / 2)² - 5 / 4.
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