The vector x in terms of the basis B is [x]B = [a].
The vector x can be written as a linear combination of the basis vectors V₁ and V₂ as shown below:
[x]B = a₁V₁ + a₂V₂
Since B is a basis of R², every vector in R² can be represented as a linear combination of V₁ and V₂.
Using the coordinates of x with respect to B, we have:[x]B = a₁V₁ + a₂V₂ = [8; 3]
We can represent V₁ and V₂ in terms of the standard basis of R² as follows:V₁ = [1; 0] and V₂ = [0; 1]
Therefore,[x]B = a₁[1; 0] + a₂[0; 1] = [a₁; a₂]Hence, we can conclude that a = [8; 3] with respect to the standard basis of R².
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A sequence , satisfies the recurrence relation with
initial
conditions and . Find an explicit formula for the sequence.
+ k2 3) A sequence a,,a,,a z ..., satisfies the recurrence relation ax = 2x-1 + 2ax-2 with initial conditions a, = 2 and a = 7. Find an explicit formula for the sequence.
The explicit formula for the sequence [tex]\(a_n\)[/tex] is:
[tex]\(a_n = \begin{cases} 4n + 3 & \text{if } n \text{ is even} \\ 4n - 2 & \text{if } n \text{ is odd} \end{cases}\)[/tex]
To find an explicit formula for the sequence [tex]\(a_n\)[/tex] that satisfies the recurrence relation [tex]\(a_n = 2n-1 + 2a_{n-2}\)[/tex] with initial conditions [tex]\(a_1 = 2\)[/tex] and [tex]\(a_2 = 7\)[/tex], we can proceed as follows:
First, let's examine the first few terms of the sequence:
[tex]\(a_1 = 2\)\\\(a_2 = 7\)\\\(a_3 = 2(3) - 1 + 2a_1 = 5 + 2(2) = 9\)\\\(a_4 = 2(4) - 1 + 2a_2 = 8 + 2(7) = 22\)\\\(a_5 = 2(5) - 1 + 2a_3 = 9 + 2(9) = 27\)\\[/tex]
We can observe that the even-indexed terms [tex]\(a_2, a_4, a_6, \ldots\)[/tex] are increasing by a factor of 2, while the odd-indexed terms [tex]\(a_1, a_3, a_5, \ldots\)[/tex] are increasing by a factor of 3. This pattern suggests that we can split the sequence into two separate sequences:
For even-indexed terms:
[tex]\(b_n = a_{2n}\)[/tex]
For odd-indexed terms:
[tex]\(c_n = a_{2n-1}\)[/tex]
Let's find explicit formulas for both [tex](\(b_n\))[/tex] and [tex](\(c_n\))[/tex]:
1. Even-indexed terms [tex](\(b_n\))[/tex]:
The recurrence relation becomes:
[tex]\(b_n = 2(2n) - 1 + 2b_{n-1}\)[/tex]
To simplify the formula, let's rewrite [tex]\(b_n\)[/tex] as [tex]\(b_{n+1}\)[/tex] (i.e., shifting the index by 1):
[tex]\(b_{n+1} = 2(2n + 2) - 1 + 2b_{n}\)[/tex]
Subtracting the two equations, we get:
[tex]\(b_{n+1} - b_n = 4\)[/tex]
This is a simple arithmetic progression with a common difference of 4. To find an explicit formula for [tex]\(b_n\)[/tex], we can use the formula for the nth term of an arithmetic progression:
[tex]\(b_n = b_1 + (n - 1) \cdot \text{{common difference}}\)[/tex]
Substituting [tex]\(b_1 = a_2 = 7\)[/tex] and the common difference of 4, we have:
[tex]\(b_n = 7 + (n - 1) \cdot 4 = 4n + 3\)[/tex]
2. Odd-indexed terms [tex](\(c_n\))[/tex]:
The recurrence relation becomes:
[tex]\(c_n = 2(2n-1) - 1 + 2c_{n-1}\)[/tex]
Similar to before, let's rewrite [tex]\(c_n\)[/tex] as [tex]\(c_{n+1}\)[/tex]:
[tex]\(c_{n+1} = 2(2n + 1) - 1 + 2c_{n}\)[/tex]
Subtracting the two equations, we get:
[tex]\(c_{n+1} - c_n = 4\)[/tex]
Again, this is an arithmetic progression with a common difference of 4. Applying the formula for the nth term of an arithmetic progression:
[tex]\(c_n = c_1 + (n - 1) \cdot \text{{common difference}}\)[/tex]
Substituting [tex]\(c_1 = a_1 = 2\)[/tex] and the common difference of 4, we have:
[tex]\(c_n = 2 + (n - 1) \cdot 4 = 4n-2[/tex]
1) [tex]\cdot 4 = 4n - 2\)[/tex]
Now that we have explicit formulas for both [tex]\(b_n\)[/tex] and [tex]\(c_n\)[/tex], we can combine them to obtain the explicit formula for the original sequence [tex]\(a_n\)[/tex]:
For even-indexed terms, [tex]\(a_{2n} = b_n = 4n + 3\)[/tex]
For odd-indexed terms, [tex]\(a_{2n-1} = c_n = 4n - 2\)[/tex]
Therefore, the explicit formula for the sequence [tex]\(a_n\)[/tex] is:
[tex]\(a_n = \begin{cases} 4n + 3 & \text{if } n \text{ is even} \\ 4n - 2 & \text{if } n \text{ is odd} \end{cases}\)[/tex]
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PLEASE HELP FAST WILL GIVE BRAINLIEST
Answer:
The answer is 25 degree because there is ( 8x-1)
Arrange the following fraction from least to greatest 2/3, 5/6, 3/5
What did you do to arrange the fraction from least to greatest?
Answer:
2/3 and 3/5 is same, then 5/6
Step-by-step explanation:
you can convert the fractions to decimals to find their value and then arrange them from least to the greatest.
Answer:
3/5, 2/3, 5/6 [From Least to Greatest]
Step-by-step explanation:
First you're going to want to know which one is "the bigger piece of pie".
I made a few drawing and look at the pictures (Just in case you have a different opinion from my answer)
Because of the Central Limit Theorem, the normal distribution is also a good approximation for the Poisson distribution. For a draw from a Poisson with parameter 1 = 37, what is the theoretical mean?
The theoretical mean for a draw from a Poisson distribution with parameter λ is equal to λ itself. In this case, λ = 37, so the theoretical mean is also 37.
Explanation:
The Poisson distribution is commonly used to model the number of events occurring within a fixed interval of time or space, when these events occur with a known average rate λ. The probability mass function of the Poisson distribution is given by P(X=k) = (e^(-λ) * λ^k) / k!, where X represents the random variable representing the number of events and k is the observed value.
The Central Limit Theorem states that when independent random variables are added, their sum tends toward a normal distribution, regardless of the shape of the original distribution. For a Poisson distribution, as the parameter λ increases, the distribution becomes more symmetric and bell-shaped, resembling a normal distribution.
Since the mean of a Poisson distribution is equal to its parameter λ, the theoretical mean for a draw from a Poisson distribution with parameter 1 = 37 is 37. This means that, on average, 37 events are expected to occur within the given interval.
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PLEASE HELP IF I DONT PASS THIS TEST I FAIL AND I DON'T UNDERSTAND IT AND I AM ON THE VERGE OF MENTAL BREAKDOWN
Answer: One of the ways you can do all this is by zearn or by listening by your teacher.
Answer:
1 e
2 a
3 b
4 c
5 d
Step-by-step explanation:
i did the answers to the first question i dont know the rest sorry
a is (4,15) and b is (8,1) what is the midpoint of AB?
Answer:
(6,8)
Step-by-step explanation:
midpoint=(x1+x2)÷2,(y1+y2)÷2
a(4,15) b(8,1)
x=4+8=12÷2=6
y=15+1=16÷2=8
Answer=(6,8)
I just need the answer if you don’t have a real answer but want to say something just comment it
Answer:
80??
Step-by-step explanation:
In order to check if blood pressure measurements change if one is sitting or standing, a study was conducted where systolic blood pressure of 35 patients were recorded while in sitting position and then again while standing. The comparison of systolic blood pressure in the two positions is an example of testing the difference between: a-Two means from independent populations b-Two population proportions c-Matched pairs from two dependent populations d-All of the above options are equally viable testing methods
The comparison of systolic blood pressure in the sitting and standing positions is an example of testing the difference between matched pairs from two dependent populations.
The scenario described involves measuring the systolic blood pressure of the same set of patients in two different positions (sitting and standing). This creates a dependency between the measurements because each patient serves as their own control. In this case, the appropriate statistical test would be a paired t-test or a related test for dependent samples.
Two means from independent populations: This option would be suitable if the measurements were taken from two different groups of patients who were independent of each other, but in this case, the same individuals were measured in both positions. Two population proportions: This option would be applicable if the data involved proportions or categorical variables, rather than continuous measurements like blood pressure.
Matched pairs from two dependent populations: This option accurately represents the scenario described, as the measurements were taken from the same individuals in both positions, making them dependent on each other.
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Which of the following inequalities has the graphed solution below?
zzGroup of answer choices
x − 1 ≥ 0
x − 1 ≤ 0
x + 1 ≥ 0
x + 1 ≤ 0
sketch the strophoid shown below. r = sec() − 2 cos(), − 2 < < 2
The strophoid is a curve represented by the polar equation r = sec(θ) − 2cos(θ), where -2 < θ < 2. In Cartesian coordinates, the strophoid equation can be written as (x^2 + y^2)^2 = 4y^2(x + 2).
The strophoid has a unique shape characterized by its looped structure.
The strophoid is symmetric with respect to the y-axis, as changing θ to -θ gives the same value of r. It has two branches that intersect at the origin (0, 0). As θ increases from -2 to 2, the curve starts from the rightmost point of the loop, extends to the left, and then returns back to the rightmost point.
The loop of the strophoid is created by the interplay of the secant function, which stretches the curve away from the origin, and the cosine function, which pulls it towards the origin. The strophoid exhibits interesting geometric properties and is often used in mathematical modeling and visualization.
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For every six dollars that Jamal saves in his account, his brother saves eight dollars in his account.
If Jamal has $24.00 dollars in his account, how much money does his brother have in his account?
Answer:
jamel brother has 48.00 dollors
Step-by-step explanation:
Answer: He has 32$ 24 divided by 6 is 4 multiply 4 by 8 and you get 32
6=2(y+2) i need help
Answer:
y=1
Step-by-step explanation:
6=2(y+2)
6=2y+4
2=2y
y=1
Answer:
y=1
Step-by-step explanation:
PLEASE HELP I HAVE 5 MINUTES TO DO THIS AND I HAVE NO CLUE HOW
WILL MARK BRAINLIEST!!
A faraway planet is populated by creatures called Jolos. All Jolos are either
green or purple and either one-headed or two-headed.
Balan, who lives on this planet, does a survey and finds that her colony of 500
contains 100 green, one-headed Jolos: 125 purple, two-headed Jolos; and
270 one headed Jolos.
Answer:
Option B
Step-by-step explanation:
We have to complete the table given in the question,
One headed Two headed Total
Green 100 230 - 125 = 105 105 + 100 = 205
Purple 270 - 100 = 170 125 170 + 125 = 295
Total 270 500 - 270 = 230 500
By analyzing the given table,
Number of green Jolos in Balan's colony = Total of one headed green Jolos and Two headed green Jolos
= 205
Therefore, number of green Jolos in Balan's colony are 205.
Option B will be answer.
Answer:
When you put together the whole chart you will see the total is 205.
Plz help ASAP !!!!! Plzzz
Answer:
The second one
Step-by-step explanation:
She started with x dollars and then used 8 dollars to buy a football game ticket, so x-8. Then, she is left with 56 dollars, so x-8=56. Therefore, the second story represents the equation.
Find the length of side x in simplest radical form with a rational denominator.
Answer:
since it is Right angled isosceles triangle it's base side are equal
by using Pythagoras law
x²+x²=1²
2x²=1
x=√{1/2}or 0.707 or 0.71
HELP I NEED HELP ASAP
HELP I NEED HELP ASAP HELP I NEED HELP ASAP
HELP I NEED HELP ASAP HELP I NEED HELP ASAP
HELP I NEED HELP ASAP
Answer:
B
Step-by-step explanation:
Sarah's house is not shown in graph
Evaluate x-2 for x=-3
Answer:
-5
Step-by-step explanation:
Rewrite x - 2 as -3 - 2, which comes out to -5.
Mohammed is x years old.
Holly is 3 years older than Mohamed.
Karen is twice as old as Mohamed.
The total of their ages is 51.
How old is Mohamed?
Step-by-step explanation:
Mohammed age = x
Holly age = x + 3
Karen age = 2x
given,
[tex]x + (x + 3) + 2x = 51 \\ x + x + 3 + 2x = 51 \\ 4x + 3 = 51 \\ 4x = 51 - 3 \\ 4x = 48 \\ x = 48 \div 4 \\ = 12[/tex]
cos80°.cos10°-sin80°.sin10°
Step-by-step explanation:
The answer will be zero
here are the steps:
cos(90-10)xcos(90-80)-sin(90-10)xsin(90-10)=
(cos10xcos10)-(sin80xsin80)=
0.965111-0.965111=0
have a good day :)
I hope it will benefit you.
Factor 2x^2 - 3x = 2x - 2............Please?
Answer:
X = 1/2, 2
Step-by-step explanation:
= (2x-1)(x-2)
Answer:
OK I DON'T KNOW IF THIS IS IT BUT THE ANSWER IS POSSIBLY 0??? AND IF I DOESN'T WORK I'M SORRY
Step-by-step explanation:
Find the perimeter of the triangle.
A) 20
B) 51
C) 12 + 74
D) 12 + 47
Luis rolled a number cube 60 times. He rolled the number 6 four times. Which is most likely the cause of the discrepancy between Luis’s experimental outcome and the predicted outcome?
PLSS HELP IMMEDIATELY!!! i’ll give brainiest if u don’t leave a link!
Answer:
They have air-filled pockets in their leaves
Step-by-step explanation:
Can anyone help find x?
Answer:
119
Step-by-step explanation:
Answer:
x= 61
Step-by-step explanation:
i think
If AABC = ADEC,
ZB = 44º and ZE = 4x
A
B
С
E
x = [?]
Answer:
The angle at B is the same as the angle at E so equate them to each other to find x
2x+4=40°
2x=40-4
2x=36
x=36/2=18
Step-by-step explanation:
Hope this is helpful! stay safe and God Bless:)))
PLEASE HELP THIS IS TIMED!!
Answer:
(D) 3/10
Step-by-step explanation:
So its split into 10 different rates each and its on the third split So 3/10.
HELP NO LINKS JUST ANSWER DUE TODAY
I need the length of DB and Measure of angle C in degrees!!!!!
Answer:
DB = 10
m∡C = 106°
Step-by-step explanation:
DE = EB
20x - 8 = 16x + 12
4x = 20
x = 5
DB = 5 doubled, or 10
m∡A + m∡D = 180
3y + 7 + 2y + 8 = 180
5y + 15 = 180
5y = 165
y = 33
m∡A = m∡C
m∡A = 3(33)+7 = 106°
m∡C = 106° also
Multiply and combine like terms. Use^ for exponents. (3x+1)(2x^2 -9x+5)
Answer:
[tex]6x^{3} -25x^{2} +6x+5[/tex]
Step-by-step explanation: