if S_1=1,S_2=8 and S_n=S_n-1+2S_n-2 whenever n≥2. Show that S_n=3⋅2n−1+2(−1)n for all n≥1.
Answers
You can try to show this by induction:
• According to the given closed form, we have \(S_1=3\times2^{1-1}+2(-1)^1=3-2=1\), which agrees with the initial value S₁ = 1.
• Assume the closed form is correct for all n up to n = k. In particular, we assume
\(S_{k-1}=3\times2^{(k-1)-1}+2(-1)^{k-1}=3\times2^{k-2}+2(-1)^{k-1}\)
and
\(S_k=3\times2^{k-1}+2(-1)^k\)
We want to then use this assumption to show the closed form is correct for n = k + 1, or
\(S_{k+1}=3\times2^{(k+1)-1}+2(-1)^{k+1}=3\times2^k+2(-1)^{k+1}\)
From the given recurrence, we know
\(S_{k+1}=S_k+2S_{k-1}\)
so that
\(S_{k+1}=3\times2^{k-1}+2(-1)^k + 2\left(3\times2^{k-2}+2(-1)^{k-1}\right)\)
\(S_{k+1}=3\times2^{k-1}+2(-1)^k + 3\times2^{k-1}+4(-1)^{k-1}\)
\(S_{k+1}=2\times3\times2^{k-1}+(-1)^k\left(2+4(-1)^{-1}\right)\)
\(S_{k+1}=3\times2^k-2(-1)^k\)
\(S_{k+1}=3\times2^k+2(-1)(-1)^k\)
\(\boxed{S_{k+1}=3\times2^k+2(-1)^{k+1}}\)
which is what we needed. QED
Related Questions
PLZ HELP FAST
Enter an equation in standard form to model the linear situation.
A pool that is being drained contained 19,000 gallons of water. After 2 hours, 14,500 gallons of water remain.
Answers
Answer:
Step-by-step explanation:
it is losing 4 500 every 2 hours. so 2 250 every hour. It will go like 19 000
- 14 500 - 10 000 - 5 500 - 1 000. Then it will lose the rest in less than 30 minutes
Can someone explain please and give me the answer
Answers
Answer:
F. y = .25x + 3
Step-by-step explanation:
use the slope equation m = y2-y1/x2-x1 with the points. so for this problem it would look like: m = 6-2/12-(-4) which is 1/4 (.25).
For the y-intercept plug what you already know (slope and a set of points) into the slope-intercept formula: 2 = .25(-4) + b and solve.
2 = -1 +b
+1 ( add the one)
b = 3 and that's the y-intercept
What is the equation of the following line? Be sure to scroll down first to see all answer options.
A.
y = - x
B.
y = -2x
C.
y = 2x
D.
y = x
E.
y = -4x
F.
y = - x
Answers
Answer:
The answer is option FStep-by-step explanation:
Equation of a line is y = mx + c
where
m is the slope
c is the y intercept
To calculate the equation of the line first find the slope
Slope of the line using points
(0 , 0) and (4 , -2) is
\(m = \frac{ - 2 - 0}{4 - 0} = \frac{ - 2}{4} = - \frac{1}{2} \)
Now use the formula
y - y1 = m(x - x1) to find the equation of the line using any of the points
Using point (0,0)
That's
\(y - 0 = - \frac{ 1}{2} (x - 0)\)
The final answer is
\(y = - \frac{1}{2} x\)
Hope this helps you
Answer:
F
Step-by-step explanation:
4x ^ 2 + 300 = 1, 596 help solve for the x please show work I will mark brainliest
Answers
Answer:
x=18 OR x=-18
Step-by-step explanation:
\(4x^2=1296\\x^2=324\\x=+-sqrt(234)\\x=18 OR x=-18\)
A student takes a subway to a public library. The table shows the distance d (in miles) the student travels in t minutes. Determine whether the data can be modeled by a linear, exponential, or a quadratic function and then select a function rule to model the situation.
t
d
1
0.83
2
1.66
3
2.49
4
3.32
5
4.15
Answers
The linear function that models the situation is given as follows:
d = 0.85t.
What is a proportional relationship?A proportional relationship is a relationship in which a constant ratio between the output variable and the input variable is present.
The equation that defines the proportional relationship is a linear function with slope k and intercept zero given as follows:
y = kx.
The slope k is the constant of proportionality, representing the increase or decrease in the output variable y when the constant variable x is increased by one.
The constant ratio for this problem is given as follows:
k = 0.83, as each division of d by t has a result of 0.83.
Hence the equation is given as follows:
d = 0.83t.
A similar problem, also featuring proportional relationships, is presented at https://brainly.com/question/7723640
#SPJ1
84.4% of what number is 19.412?
Answers
Answer: 22.4402
Step-by-step explanation:
Answer:
84.4% of 23 is 19.412.
Step-by-step explanation:
Let the unknown number be x.
Now, 84.4% of x = 19.412.
∴ (84.4 ÷ 100) × x = 19.412.
By simplifying the equation,
x = (19.412 × 100) ÷ 84.4
x = 1941.2 ÷ 84.4
∴ x = 23
Thus, 84.4% of 23 is 19.412.
To learn more about Percentages,
refer to the following question!
https://brainly.com/question/3079428
Which expression is equivalent to StartRoot 8 x Superscript 7 Baseline y Superscript 8 Baseline EndRoot? Assume x greater-than-or-equal-to 0.
x y squared StartRoot 8 x cubed EndRoot
2 x cubed y cubed StartRoot x y squared EndRoot
2 x cubed y Superscript 4 Baseline StartRoot 2 x EndRoot
4 x cubed y Superscript 4 Baseline StartRoot x EndRoot
Answers
The expression that is equivalent to StartRoot \(8 x^7 y^8\) EndRoot is (\(2 x^3 y^4\) StartRoot 2 x EndRoot)^2.
To understand why this is the case, let's break down each expression and simplify them step by step:
StartRoot \(8 x^7 y^8\) EndRoot:
We can rewrite 8 as \(2^3\), and since the square root can be split over multiplication, we have StartRoot \((2^3) x^7 y^8\) EndRoot. Applying the exponent rule for square roots, we get StartRoot \(2^3\) EndRoot StartRoot \(x^7\) EndRoot StartRoot \(y^8\) EndRoot.
Simplifying further, we have 2 StartRoot \(2 x^3 y^4\) EndRoot StartRoot \(2^2\) EndRoot StartRoot \(x^2\) EndRoot StartRoot \(y^4\) EndRoot. Finally, we obtain 2 \(x^3 y^4\) StartRoot 2 x EndRoot, which is the expression in question.
(\(2 x y^2\) StartRoot 8 x^3 EndRoot)^2:
Expanding the expression inside the parentheses, we have \(2 x y^2\)StartRoot \((2^3) x^3\) EndRoot. Applying the exponent rule for square roots, we get \(2 x y^2\) StartRoot \(2^3\) EndRoot StartRoot \(x^3\) EndRoot.
Simplifying further, we have \(2 x y^2\) StartRoot 2 x EndRoot. Squaring the entire expression, we obtain (\(2 x y^2\) StartRoot 2 x EndRoot)^2.
Therefore, the expression (\(2 x^3 y^4\) StartRoot 2 x EndRoot)^2 is equivalent to StartRoot \(8 x^7 y^8\) EndRoot.
For more such questions on expression
https://brainly.com/question/1859113
#SPJ8
matematica ayuda xfis
Answers
Answer:
en que te ayudo
Step-by-step explanation:
2log12 3 + 4log12 2 simplifying
Answers
Hello, I was happy to solve the problem. If you find a bug, post in the comments or click on the report, I will see it and try to fix it as soon as possible.
The answer to this problem: 2
can you solve by steps and show me the focus equation please?
Answers
the length of the rectangular base is 2.5 times of width
let the width is b = x
then length is l = 2.5x
the volume of the box is 650 cm^2
let the height is h
so volume is
l x b x h = 650
(2.5x) (x) h = 650
h = 650/2.5x^2650
now the area of the base is
\(\begin{gathered} A=x\times2.5x \\ A=2.5x^2 \end{gathered}\)the cost of the base is 6 cents/ cm^2
now the total cost is,
6 (2.5x^2) = 15x^2
area of the sides is
A= 4 (l x h)
\(\begin{gathered} A=4\times2.5x\times h \\ A=10x\times h \end{gathered}\)cost of the side is 6 cents/ cm^2
so total cost os sides are (10x)h
so total cost of the box is
15x^2 + 10xh
now, put the value of h
\(\begin{gathered} =15x^2+10x\times\frac{650}{2.5x^2} \\ =15x^2+\frac{4\times650}{x} \end{gathered}\)for minimizing the cost
differentiate the above expression
30x - 2600/x^2 =0
from here x = 4.42
now the dimension are
b= 4.42
l = 2.5 x 4.42 = 11.05
h = 13.30
What is the total number of different 9-letter arrangements that can
be formed using the letters in the word Projector?
Answers
Find the value of 5⁰ X 7⁰ X 3⁰
plz help asap
Answers
Answer:
5⁰ × 7⁰ × 3⁰ = 1
Step-by-step explanation:
5⁰ × 7⁰ × 3⁰ = ?
5⁰ = 1
7⁰ = 1
3⁰ = 1
1 × 1 × 1 = 1
what is the equivalent fraction of 13%
Answers
find the cost of 200 shares of stock that sell for 11.3 per share
Answers
in a quadrilateral the angles are degree x plus 10 degree x plus 20 degree and c plus 30 degree find angles
Answers
Answer:
\(75; 85; 95; 105\)
Step-by-step explanation:
Given
\(Angles: x, x + 10, x + 20, x + 30\)
Required
Find the angles
Angles in a \(quadrilateral \add\ up\ to\ 360\)
So:
\(x+ x + 10+ x + 20+ x + 30 = 360\)
Collect like terms
\(x+ x + x + x = 360 - 10 - 20 - 30\)
\(4x = 300\)
Solve for x
\(x = \frac{300}{4}\)
\(x = 75\)
So, the angles are:
\(x = 75\)
\(x +10= 75 + 10 = 85\)
\(x +20= 75 + 20 = 95\)
\(x +30= 75 + 30 = 105\)
uppose that a single die with 8 sides (numbered 1, 2, 3, ... , 8) is rolled twice. What is the probability that the sum of the two rolls equals 5
Answers
Answer:
1/16 = 0.0625 = 6.25% probability that the sum of the two rolls equals 5.
Step-by-step explanation:
A probability is the number of desired outcomes divided by the number of total outcomes.
Die with 8 sides
This means that the total number of outcomes is \(T = 8^2 = 64\)
Outcomes in which the sum of the two rolls is 5:
(1,4), (4,1), (2,3), (3,2)
4 desired outcomes, so \(D = 4\)
What is the probability that the sum of the two rolls equals 5?
\(p = \frac{D}{T} = \frac{4}{64} = 0.0625\)
1/16 = 0.0625 = 6.25% probability that the sum of the two rolls equals 5.
The probability that the sum of the two rolls equals 5 is 1/16
What is probabilityProbability is the likelihood or chance that an event will occur.
Probability = Expected outcome/Total outcome
Given a single die with 8 sides (numbered 1, 2, 3, ... , 8) is rolled twice, the total outcome will be:
Total outcome = 8^2
Total outcome = 64
For the sum of two rolls, the expected outcome will be:
E = {(1, 4), (4, 1), (3, 2), (2, 3)}
n(E) = 4
Pr(sum of the two rolls is 5) = 4/64
Pr(sum of the two rolls is 5) = 1/16
Hence the probability that the sum of the two rolls equals 5 is 1/16
Learn more on probability here; https://brainly.com/question/24756209
In the grid below, list something you know about U.S. history. Next to that item, list a question you have about that time in history. An example has been provided for you.
Answers
In the write up below, i have explained what i know about U.S. history as regards the dark age.
What is the history about USA?A little about the history maybe in the Medieval history which is also known as Middle Age.
The word Medieval comes from a Latin word which means Middle age. This period is the known as middle age because it is present in the middle of ancient Greek and Age of Discovery. It started in 5th century when the Roman empire fall and ended in 15th century with the Age of Discovery. the early years of Medieval period is also called Dark Age.
Apart from a damp corner of Canada, there simply was no Middle Ages in North America. This is because “the Middle Ages” is not just a temporal category, but a geographically- and culturally-specific one.
The idea of the Middle Ages was conceived by Italian humanists in the fifteenth century as a way of setting themselves apart from what came before. They wanted to distance themselves from what (they argued) was a time of darkness that engulfed the gap between the glories of the Roman Empire and their glittering present in the fifteenth century.
Read more about U.S History at; https://brainly.com/question/9515546
#SPJ1
Point D is in the interior of
Answers
Answer:
Angle ABC,
Step-by-step explanation:
Please give more detail to your question, this doesn't help at all...
m∠ABD + m∠DBC = m∠ABC
Find the value of the expression below if x=10. x2−2(x+5)
PLZ HELP GIVING BRANLIEST
Answers
-10
Step-by-step explanation:
x = 10
x2 - 2(x + 5)
10(2) - 2(10+5)
20 - 2(15)
20 - 30
-10
i dont know how to do this, help
Answers
Answer:
Gerald should expect the spinner to land on 3, 126 times.
Step-by-step explanation:
Because we don't know what's actually going to happen, we can determine the theoretical probability by multiplying 1/5 by 630 to get 126.
Answer:
126
Step-by-step explanation:
Given that all of the spaces are equally sized, each space has an equal chance of being landed on by the spinner. As there are 5 spaces, the probability of landing on any one space is 1/5. This means that one fifth of the number of spins are expected to land on 3.
630 × 1/5 = 126
Which best describes how the graph of g(x) = square root of 25/9 x relates to the graph of the parent function, f(x) = square root of x?
The graph of g(x) is shrunk vertically by a factor of 5/3.
The graph of g(x) is stretched vertically by a factor of 5/3.
The graph of g(x) is shrunk vertically by a factor of 25/9.
The graph of g(x) is stretched vertically by a factor of 25/9.
Answers
Answer:
The graph of g(x) is stretched vertically by a factor of 5/3.
Step-by-step explanation:
f(x) = √x
g(x) = √25/9 x = 5/3 √x
5/3 > 1 .. vertical stretch by a factor of 5/3
Please help me !! I need help asap
Answers
In given fractions 8/9 is 1/36 large than 31/36
Comparing Fractions:Finding the larger and smallest fraction in between two or more fractions is known as comparing fractions.
We need to change fractions with unlike denominators into fractions with similar denominators in order to compare them. For this, we will find the denominators' Least Common Multiple (LCM).
If the denominators are the same then it is easy to compare the fractions.
Here we have
31/36 and 8/9
To find the largest fraction convert both denominators into the same
Here LCM(36, 9) = 36
Now multiply both numerator and denominators of fractions with a number to change the denominators
=> \(\frac{31}{36} \times \frac{1}{1} = \frac{31}{36}\)
=> \(\frac{8}{9} \times \frac{4}{4} = \frac{32}{36}\)
From the above calculations given fractions are 31/36 and 32/36
Difference = \(\frac{32}{36} - \frac{31}{36} = \frac{1}{36}\)
Therefore,
In given fractions 8/9 is 1/36 large than 31/36
Learn more about Fractions at
https://brainly.com/question/14503941
#SPJ1
A fair number cube with 1, 2, 3, 4, 5 and 6 on its faces is rolled once. The dot on the number line below represents the probability of an event.
Answers
The probability of rolling a specific number is 1/6 or approximately 0.1667.
When rolling a fair number cube with faces numbered 1 through 6, each face has an equal chance of landing face-up. This means the probability of rolling any specific number is 1 out of the total number of faces, which is 6. Therefore, the probability of rolling a specific number is 1/6 or approximately 0.1667.
The dot on the number line represents the probability of an event occurring when rolling the number cube. To determine what this event could be, consider the possible outcomes when rolling the cube and compare their probabilities to the dot's position on the number line.
For example, if the dot is located at 1/6 on the number line, this could represent the probability of rolling a single specific number, such as rolling a 3. If the dot is located at 1/2 or 0.5, it could represent the probability of rolling an even number (2, 4, or 6) since there are three even numbers out of the six total faces, making the probability 3/6 or 1/2.
In summary, the dot on the number line represents the probability of an event occurring when rolling a fair number cube. To identify the event, compare the dot's position on the number line with the probabilities of different outcomes resulting from rolling the cube.
Know more about Probability here:
https://brainly.com/question/13604758
#SPJ11
Which equation is quadratic in form?
2(x + 5)2 + 8x + 5 + 6 = 0
x6 + 6x4 + 8 = 0
7x6 + 36x3 + 5 = 0
4x9 + 20x3 + 25 = 0
Answers
Answer:
I think it's the first one... but I'm not pretty sure.
Answer:
C
Step-by-step explanation:
edge 2020
please help!!!!!!!!!!!
Answers
Answer:
pumpkin: 5
witch: 10
leaf: 50
bat: 20
ghost: 60
Answer:
Bat: 20
Pumpkin: 5
Ghost: 60
Leaf: 50
Witch: 10
Salve this proportion 15/35=g/7
Answers
1. Determine the discriminant of 3x²-5x+4=0.
A: -73
B: -60
C: -23
D: 2
Answers
How many classes would Gina need to take for the total cost to be the same at both dance studios
Answers
Answer:
How many classes is Gina taking?
Answer:
10
Step-by-step explanation:
right on edge
Please answer this correctly without making mistakes
Answers
Answer:
The answer is $0.07 cents per inch
Step-by-step explanation:
If you take 6 feet into inches (72)
and divide the total cost by the amount of inches (5.04 / 72)
You get the answer
The Thief Wore Size Six Shoes
Answers
Answer:
im not going to do all of them, but i will do one just to help you get on track!
1. Y=20
Step-by-step explanation:
3Y-58=2
+58 +58
\(\frac{3Y}{3}\) = \(\frac{60}{3}\)
Y=20
