Check the forward differences of the sequence.
If [tex]\{a_n\} = \{2,6,12,20,30,42,\ldots\}[/tex], then let [tex]\{b_n\}[/tex] be the sequence of first-order differences of [tex]\{a_n\}[/tex]. That is, for n ≥ 1,
[tex]b_n = a_{n+1} - a_n[/tex]
so that [tex]\{b_n\} = \{4, 6, 8, 10, 12, \ldots\}[/tex].
Let [tex]\{c_n\}[/tex] be the sequence of differences of [tex]\{b_n\}[/tex],
[tex]c_n = b_{n+1} - b_n[/tex]
and we see that this is a constant sequence, [tex]\{c_n\} = \{2, 2, 2, 2, \ldots\}[/tex]. In other words, [tex]\{b_n\}[/tex] is an arithmetic sequence with common difference between terms of 2. That is,
[tex]2 = b_{n+1} - b_n \implies b_{n+1} = b_n + 2[/tex]
and we can solve for [tex]b_n[/tex] in terms of [tex]b_1=4[/tex]:
[tex]b_{n+1} = b_n + 2[/tex]
[tex]b_{n+1} = (b_{n-1}+2) + 2 = b_{n-1} + 2\times2[/tex]
[tex]b_{n+1} = (b_{n-2}+2) + 2\times2 = b_{n-2} + 3\times2[/tex]
and so on down to
[tex]b_{n+1} = b_1 + 2n \implies b_{n+1} = 2n + 4 \implies b_n = 2(n-1)+4 = 2(n + 1)[/tex]
We solve for [tex]a_n[/tex] in the same way.
[tex]2(n+1) = a_{n+1} - a_n \implies a_{n+1} = a_n + 2(n + 1)[/tex]
Then
[tex]a_{n+1} = (a_{n-1} + 2n) + 2(n+1) \\ ~~~~~~~= a_{n-1} + 2 ((n+1) + n)[/tex]
[tex]a_{n+1} = (a_{n-2} + 2(n-1)) + 2((n+1)+n) \\ ~~~~~~~ = a_{n-2} + 2 ((n+1) + n + (n-1))[/tex]
[tex]a_{n+1} = (a_{n-3} + 2(n-2)) + 2((n+1)+n+(n-1)) \\ ~~~~~~~= a_{n-3} + 2 ((n+1) + n + (n-1) + (n-2))[/tex]
and so on down to
[tex]a_{n+1} = a_1 + 2 \displaystyle \sum_{k=2}^{n+1} k = 2 + 2 \times \frac{n(n+3)}2[/tex]
[tex]\implies a_{n+1} = n^2 + 3n + 2 \implies \boxed{a_n = n^2 + n}[/tex]
help. I can't do maths again
Answer:
x ≈ 14.4 cm
Step-by-step explanation:
the third angle of the triangle = 180° - 40° - 46° = 94°
Using the Sine rule in the triangle
[tex]\frac{x}{sin46}[/tex] = [tex]\frac{20}{sin94}[/tex] ( cross- multiply )
x × sin94° = 20 × sin46° ( divide both sides by sin94° )
x = [tex]\frac{20sin46}{sin94}[/tex] ≈ 14.4 cm ( to the nearest tenth )
Answer:
14 cm (rounded to nearest whole number)
Step-by-step explanation:
Law of Sine:
[tex]\displaystyle \large{\dfrac{\sin A}{a} = \dfrac{\sin B}{b} = \dfrac{\sin C}{c} = 2R}[/tex]
a,b,c are side lengths.R is radius so 2R is diameter.A,B,C are angles.Given angles are:
40°, 46°Definition of Euclidean Triangle:
Sum of three interior angles equals 180°Find another angle:
40°+46°+B = 180°86+B = 180B = 180-86B = 94°So another angle is 94°.
To find:
Value of xDetermine:
A = 46°a = x cmB = 94°b = 20 cmTherefore:
[tex]\displaystyle \large{\dfrac{\sin 46^{\circ}}{x} = \dfrac{\sin 94^{\circ}}{20}}[/tex]
Multiply both sides by 20x:
[tex]\displaystyle \large{\dfrac{\sin 46^{\circ}}{x} \cdot 20x = \dfrac{\sin 94^{\circ}}{20} \cdot 20x}\\\displaystyle \large{20\sin 46^{\circ}= x\sin 94^{\circ}}[/tex]
Divide both sides by [tex]\displaystyle \large{\sin 94^{\circ}}[/tex]:
[tex]\displaystyle \large{\dfrac{20\sin 46^{\circ}}{\sin 94^{\circ}} = \dfrac{x\sin 94^{\circ}}{\sin 94^{\circ}}}\\\displaystyle \large{\dfrac{20\sin 46^{\circ}}{\sin 94^{\circ}} = x}[/tex]
Evaluate the expression, hence:
[tex]\displaystyle \large{x = 14.42...}[/tex]
Round to nearest whole number:
[tex]\displaystyle \large{x = 14}[/tex]
Therefore, the value of x is 14 cm.
Alina drew a model of a square pyramid. The dimensions of the model are shown in the diagram. pls don't answer with links
A: 2,880 cm3
B: 480 cm3
C: 320 cm3
D: 960 cm3
Answer:
C
Step-by-step explanation:
Volume of a pyramid = 1/3 base area * height
= 1/3 (8 * 10)(12)
Does this figure have symmetry?
E
CLEAR
CHECK
Yes, because if you fold it in half, both halves match up.
Yes, because if you rotate it 90 degrees, the figure carries
onto itself.
No, because if you fold it in half, both halves do not match
up
No, because if you rotate it 180 degrees, it does not carry
onto itself.
Answer:
The Letter E has 1 line of symmetry. (Horizontal)
Step-by-step explanation:
Yes, because if you fold it in half (horizontally), both halves match up.
1. Which expression below represents
"k more than 8"?
A 8k
B 8+k
C 8-k
D 8/k
Answer:
B. 8+kStep-by-step explanation:
A variable is an undetermined number represented by a letter.
K is a variable.
The addition of more than is not a subtraction.
Therefore, the correct answer is B. 8+k.I hope this helps. Let me know if you have any questions.
Find the equation of the line shown
Answer:
[tex]y=-\frac{1}{4}x+2[/tex]
Step-by-step explanation:
We will use the point slope formula: [tex]y=mx+b[/tex]
We will start by finding the slope([tex]m[/tex]).
The formula for slope is
[tex]\frac{y_{2}-y_{1}}{x_{2}-x_{1}}[/tex]
So if we just choose 2 random points on the line:
I choose (0,2) and (4,1), we get:
[tex]\frac{1-2}{4-0} = \frac{-1}{4}[/tex]
That's our slope.
Next we need to find the y-intercept([tex]b[/tex]). We can find that by looking at the graph. It is where x=0. So in this case y=2 when x=0.
So in point-slope form the equation is:
[tex]y=-\frac{1}{4}x+2[/tex]
How to tell the difference between permutation and combination?.
Answer:
Permutation is an arrangement of things where order of arrangement matters. The position of each thing in a permutation matters. Thus, Permutation can be associated with Position.
Combination is grouping/selection of things where order does not matter. Permutation can be considered as an ordered combination.
Reflect the triangle in the x-axis
I think this might helpful
Click this link to view O*NET’s Work Activities section for Loading Machine Operators. Note that common activities are listed toward the top, and less common activities are listed toward the bottom. According to O*NET, what are some common work activities Loading Machine Operators perform? Select four options. Operating vehicles, mechanized devices, or equipment interacting with computers scheduling work and activities identifying objects, actions, and events inspecting equipment, structures, or material getting information
Answer:
so it is 12 45
Step-by-step explanation:
Answer:
its actually 1 4 5 6
Step-by-step explanation:
Please answer. I’ll mark brainliest
Answer:
x=48 degrees
Step-by-step explanation:
since two sides are the same, the angles of those sides are the same, so you add 66 twice and subtract from 180.
a bug population doubles every 5 days. If you start with 6 bugs, how many bugs will there be in 35 days
Answer:
768 bugs
Step-by-step explanation:
You can rewrite this problem as a function as time where the bug population is f(x), and x is the number of days since the start.
f(x)=6*2^(x/5)
Here, the 6 represents the number of bugs that you start with, the two shows that they double every day, and the /5 shows that they double every 5 days.
By plugging in 35, you get 6*2^7, which is 768.
Which expression is equivalent to startfraction 4 a 4 over 2 a endfraction times startfraction a squared over a 1 endfraction?
The expression that is equivalent to [(4a + 4)/2a] * [a²/(a + 1)] after simplification is; 2a
How to Simplify Fractions?We want to simplify the expression;
[(4a + 4)/2a] * [a²/(a + 1)]
This expression can be further reduced to;
[(4a + 4)/2] * [a/(a + 1)]
Factorizing the numerator to get;
[(4(a + 1)/2] * [a/(a + 1)]
(a + 1) will cancel out to get;
2a
Read more about Simplification of Fractions at; https://brainly.com/question/26255345
Type the correct answer in each box.
Ella rolls a dle and then flips a coin. The sample space for this compound event is represented in the table (H is heads and Tis tails). Complete
the table and the sentence beneath it.
Die
1
2.
3
4
5
6
heads
H-1
H-2
H-3
H-5
H-6
Coin
tails
T-1
T-3
T-4
T-5
The size of the sample space is
Hi
The size of the sample space would be { H-1 H-2 H-3 H-5 H-6 T-1 T-3 T-4 T-5 }.
What is the sample space for a random experiment?Sample space is the set of all possible outcome the considered random experiment can result to.
Thus, suppose we toss a coin, then tossing a coin is a random experiment, and the sample space is {'head', 'tail'}.
We ignore other outcomes like slant coin, or standing coin by assuming that coin is very thin to stand and always falls down.
Ella rolls a die and then flips a coin. The sample space for this compound event is represented in the table (H is heads and T is tails).
Die 1 2 3 4 5 6
Coin: heads H-1 H-2 H-3 H-5 H-6
tails T-1 T-3 T-4 T-5
The size of the sample space
= { H-1 H-2 H-3 H-5 H-6 T-1 T-3 T-4 T-5 }
Learn more about sample space of a random experiment here:
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Answer:
hope this works for u
Step-by-step explanation:
A car travels xkm in 3 hours. How far will it travel in 9 hours?
Answer:
3x km
Step-by-step explanation:
Pooja and Amit are playing a card game with a standard 52 card deck. Pooja will draw a card. If the card she draws is a face card (a jack, queen, or king), she will win $6. If Pooja draws an ace, she will win $4. If any other card is drawn, she will lose $2. The image below shows a standard deck of playing cards.
What is Pooja's expected value of playing this game?
The probability helps us to know the chances of an event occurring. Pooja's expected value of playing this game is 0.3077.
What is Probability?The probability helps us to know the chances of an event occurring.
[tex]\rm Probability=\dfrac{Desired\ Outcomes}{Total\ Number\ of\ outcomes\ possible}[/tex]
There are 12 face cards, therefore, the probability of the getting a face card can be written as,
[tex]\text{Probability of face card }= \dfrac{12}{52} = 0.2308[/tex]
Also, the winning prize for drawing a face card is $6. Therefore, the expected value will be,
[tex]\text{Expected Value(Face)} = \$6 \times 0.2308= $1.3846[/tex]
There are 4 ace cards, therefore, the probability of the getting an ace card can be written as,
[tex]\text{Probability of ace card }= \dfrac{4}{52} = \dfrac{1}{13} = 0.077[/tex]
Also, the winning prize for drawing an ace card is $4. Therefore, the expected value will be,
[tex]\text{Expected Value(Ace)} = \$4 \times 0.077 = $0.3077[/tex]
There are 36 normal cards, therefore, the probability of the getting a normal card can be written as,
[tex]\text{Probability of normal card }= \dfrac{36}{52} = 0.6923[/tex]
Also, the losing prize for drawing a normal card is -$2. Therefore, the expected value will be,
[tex]\text{Expected Value(Normal)} = (-\$2) \times 0.6923 = -$1.3846[/tex]
Thus, Pooja's expected value of playing this game can be written as,
[tex]\rm Expected\ Value = 1.3846 + 0.3077-1.3846 = 0.3077[/tex]
Hence, Pooja's expected value of playing this game is 0.3077.
Learn more about Probability:
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Giving 100 points
The dot plot below shows the hourly rate of some babysitters in a city:
A number line is shown from 1 dollar to 5 dollars in increments of 1 dollar for each tick mark. The horizontal axis label is Dollar per hour. There are 4 dots above 2 dollars, 2 dots above 3 dollars, 3 dots above 4 dollars, and 1 dot above 5 dollars. The title of the line plot is Babysitting Rates.
Which statement best describes the shape of the data?
It is symmetric and has a peak at $5.00.
It is not symmetric and has a peak at $2.00.
It is a cluster from $1.00 to $5.00 and has gaps.
It is a cluster from $1.00 to $5.00 and has no gaps.
Answer:
It is not symmetric and has a peak at $2.00
Step-by-step explanation:
$2.00 has 4 dots while the others have less
in order for it to be symmetric:
1 and 5 would need the same amount2 and 4 would need the same amount3 could be whateverit is also not clustered from $1.00 to $5.00 because $1.00 has no amount
There are no gaps as $1.00 has no amount
Answer:
It is not symmetric and has a peak at $2.00
Step-by-step explanation:
hope i helped you very much
at which angles are the x and y coordinate on the unit circle exactly the same? give your answer in radians and degrees
Answer:
In radian :
[tex]\frac{\pi }{4} \ \text{and}\ \frac{5\pi }{4}[/tex]
In degree :
45° and 225°
Step-by-step explanation:
the x and y coordinate on the unit circle exactly the same at the angles :
In radian :
[tex]\frac{\pi }{4} \ \text{and}\ \frac{5\pi }{4}[/tex]
In degree :
45° and 225°
At those angles :
[tex]\left( x,y\right) =\left( \frac{\sqrt{2} }{2} ,\frac{\sqrt{2} }{2} \right) \ or\ \left( x,y\right) =\left( -\frac{\sqrt{2} }{2} ,-\frac{\sqrt{2} }{2} \right)[/tex]
Who the answer so ez to find
Mee
increase 31 kg by 100%
find x. the whole question is- To what angle of depression should the security camera be adjusted in order to have lens aimed at point P on the ground?
Answer:
Hence, the security camera be adjusted in order to have the lens aimed at point. P. on the ground at an angle of depression of. x∘=30∘
Step-by-step explanation:
5 If the standard deviation of a distribution 3.2, find the varience correct to the nearest whole number
Answer:
Step-by-step explanation:
variance = (S.D)^2
=3.2^2
=10.24
=10
therefore , varience of a distribution is 10
What is 0.47 expressed as a fraction in simplest form?
(SHOW WORK)
Answer: 0.47 is equal to 47/100
Step-by-step explanation:
1.00=100 Is like pennies, if you have a hundred of them you have 1 dollar but since there’s only 47 it’s out of a hundred
Answer:
0.47 as a fraction equals 47/100
Step-by-step explanation:
Write 0.47 as
Multiply both the numerator and denominator by 10 for each digit after the decimal point.
[tex]\frac{0.47}{1}[/tex] = [tex]\frac{0.47 x 100 } {1x100}[/tex] = [tex]\frac{47}{100}[/tex]
So, 0.47 as a fraction in simpliest form equals 47/100
Which of the following is NOT a function?
A. {(3, 8), (3, 9)}
B. {(3, 6), (6, 12)}
C. {(3, 7), (7, 13)}
D. {(3, 16), (4, 31)}
Answer:
A. is NOT the function.
Step-by-step explanation:
In a function, 1 input (the 3) can not have two differnt ouputs (8 and 9) meaning that this is NOT a dunction.
I hope this helps! :)
Write the equation of the given line in slope-intercept form
Answer:
y = -3x -1
Step-by-step explanation:
Hi there!
We are given a line on a coordinate grid, with 2 marked points, (-1, 2) and (1, -4)
We can use these points to help find the equation of the line, which we can write in slope-intercept form
Slope-intercept form is given as y=mx+b, where m is the slope and b is the y intercept
First, we need to find the slope of the line
The formula for the slope calculated from 2 points is [tex]\frac{y_2-y_1}{x_2-x_1}[/tex], where [tex](x_1, y_1)[/tex] and [tex](x_2, y_2)[/tex] are points
We have everything we need to find the slope, but let's label the values of the points to help avoid confusion.
[tex]x_1=-1\\y_1=2\\x_2=1\\y_2=-4[/tex]
Now substitute these values into the formula to find the slope
m=[tex]\frac{y_2-y_1}{x_2-x_1}[/tex]
m=[tex]\frac{-4-2}{1--1}[/tex]
Simplify
m=[tex]\frac{-4-2}{1+1}[/tex]
m=[tex]\frac{-6}{2}[/tex]
Divide
m= -3
The slope of the line is -3
We can substitute this into the formula.
Here's our line so far:
y = -3x + b
Now we need to find b
As the equation passes through the points (-1, 2) and (1, -4), we can use either one to find the value of b
Taking (-1, 2) for example:
Substitute -1 as x and 2 as y.
2 = -3(-1) + b
multiply
2 = 3 + b
Subtract 3 from both sides
-1 = b
Substitute -1 as b.
y = -3x - 1
Hope this helps!
See more on this topic here: https://brainly.com/question/27402935
Which statement describes the relationship between point P and the other points in the scatter plot?
Answer:
A
Step-by-step explanation:
the graph is mostly going in an upward direction therefore both x and y are increasing.
Find the solution of this system of equations
-2y = 48 + 6x
-6x + 9y = -84
Which equation represents a line that passes through (–9, –3) and has a slope of –6? y – 9 = –6(x – 3) y 9 = –6(x 3) y – 3 = –6(x – 9) y 3 = –6(x 9)
Answer:
Y+3=-6(x+9)
Step-by-step explanation:
Got it right on usa test prep
Please show work thank you so much y’all
Answer:
a
Step-by-step explanation:
looked it up if u dondon't like the answer sorry
What is the volume of this prism? Enter your answer in the box. cm³ Length = 7 Width = 20 height = 4
Answer:
The volume is 560
Answer:
the volume of the right rectangular prism is 263.52 cubic centimeters hope it helps
6.
Use the Change of Base Formula to evaluate log base 4 20
Answer:
The answer with calculator would be 2.16096
What is the balance after 3 years on a CD with an initial investment of $1,800.00 and a 2.75% interest rate?
Answer:
$1952.62
Step-by-step explanation:
1,800(1+.0275)^3
= 1952.62
PLS HELP
What is the median of this data set?
Answer:
$3.50
Step-by-step explanation:
add up all the x's and distribute them starting from the outside until you get to the middle. if there is an odd number of x's (which there isn't), the last x to be placed will be on the spot that is the median. If there is an even number of x's, and they both land on the same spot, that spot will be the median, and if they are next to each other, the median will be between them.