[tex] \large\underline{\sf{Solution-}}[/tex]
Given series is
[tex]\rm \longmapsto\:\displaystyle\sum_{k=1}^{\infty}\rm \displaystyle\sum_{n=1}^{\infty}\rm \frac{k}{ {2}^{n + k} } [/tex]
can be further rewritten as
[tex]\rm \: = \: \:\displaystyle\sum_{k=1}^{\infty}\rm \displaystyle\sum_{n=1}^{\infty}\rm \frac{k}{ {2}^{k} . {2}^{n} } [/tex]
can be further rewritten as
[tex]\rm \: = \: \displaystyle\sum_{k=1}^{\infty}\rm \bigg[\dfrac{k}{ {2}^{k}}\displaystyle\sum_{n=1}^{\infty}\rm \dfrac{1}{ {2}^{n} } \bigg][/tex]
[tex]\rm \: = \: \displaystyle\sum_{k=1}^{\infty}\rm \dfrac{k}{ {2}^{k} }\bigg(\dfrac{1}{2} + \dfrac{1}{ {2}^{2} } + \dfrac{1}{ {2}^{3} } - - - \infty \bigg) [/tex]
So, its an infinite GP series with
[tex] \purple{\rm \longmapsto\:a = \dfrac{1}{2}}[/tex]
[tex] \purple{\rm \longmapsto\:r = \dfrac{1}{2}}[/tex]
We know,
Sum of infinite GP series with common ratio r ( - 1 < r < 1 ) and first term a is given by
[tex]\boxed{\tt{ \: \: S_ \infty = \frac{a}{1 - r} , \: \: provided \: that \: |r| < 1}}[/tex]
So, 1using this, we get
[tex]\rm \: = \: \displaystyle\sum_{k=1}^{\infty}\rm \frac{k}{ {2}^{k} }\bigg( \frac{ \frac{1}{2} }{1 - \frac{1}{2} }\bigg)[/tex]
[tex]\rm \: = \: \displaystyle\sum_{k=1}^{\infty}\rm \frac{k}{ {2}^{k} }\bigg( \frac{ \frac{1}{2} }{ \frac{2 - 1}{2} }\bigg)[/tex]
[tex]\rm \: = \: \displaystyle\sum_{k=1}^{\infty}\rm \frac{k}{ {2}^{k} }\bigg( \frac{ \frac{1}{2} }{ \frac{1}{2} }\bigg)[/tex]
[tex]\rm \: = \: \displaystyle\sum_{k=1}^{\infty}\rm \frac{k}{ {2}^{k} }[/tex]
[tex]\rm \: = \: \dfrac{1}{2} + \dfrac{2}{ {2}^{2} } + \dfrac{3}{ {2}^{3} } + \dfrac{4}{ {2}^{4} } + - - - \infty [/tex]
Let assume that
[tex]\rm \longmapsto\:S_ \infty = \: \dfrac{1}{2} + \dfrac{2}{ {2}^{2} } + \dfrac{3}{ {2}^{3} } + \dfrac{4}{ {2}^{4} } + - - - \infty [/tex]
Now, its an infinite Arithmetico Geometrico Series, So multiply by 1/2 we get
[tex]\rm \longmapsto\:\dfrac{1}{2} S_ \infty = \: \dfrac{1}{ {2}^{2} } + \dfrac{2}{ {2}^{3} } + \dfrac{3}{ {2}^{4} } + \dfrac{4}{ {2}^{5} } + - - - \infty [/tex]
On Subtracting above two equations, we get
[tex]\rm \longmapsto\:\dfrac{1}{2} S_ \infty = \: \dfrac{1}{2} + \dfrac{1}{ {2}^{2} } + \dfrac{1}{ {2}^{3} } + \dfrac{1}{ {2}^{4} } + - - - \infty [/tex]
[tex]\rm \longmapsto\:\dfrac{1}{2} S_ \infty = \: \dfrac{ \dfrac{1}{2} }{1 - \dfrac{1}{2} } [/tex]
[tex]\rm \longmapsto\:\dfrac{1}{2} S_ \infty = \: \dfrac{ \dfrac{1}{2} }{ \dfrac{1}{2} } [/tex]
[tex]\rm \longmapsto\:\dfrac{1}{2} S_ \infty = \: 1[/tex]
[tex]\bf\implies \:S_ \infty = 2[/tex]
Hence,
[tex] \\ \rm \longmapsto\:\boxed{\tt{ \: \: \: \: \displaystyle\sum_{k=1}^{\infty}\rm \displaystyle\sum_{n=1}^{\infty}\rm \frac{k}{ {2}^{n + k} } = 2 \: \: \: \: }} \\ [/tex]
PLEASE HELP ME WITH THIS ONE
Answer:
b answer
Step-by-step explanation:
hope helpful
if wrong sorry
Given the parent function f(x) = x2 describe the graph of g(x) = (3x-6)2 +3.
Answer:
For this case we have the following functions transformation:
Vertical expansions:
To graph y = a * f (x)
If a> 1, the graph of y = f (x) is expanded vertically by a factor a.
f1 (x) = (3x) ^ 2
Horizontal translations
Suppose that h> 0
To graph y = f (x-h), move the graph of h units to the right.
f2 (x) = (3x-6) ^ 2
Vertical translations
Suppose that k> 0
To graph y = f (x) + k, move the graph of k units up.
g (x) = (3x-6) ^ 2 + 3
Answer:
expanded horizontally by a factor of 3, horizontal shift rith 6, vertical shift up 3
Step-by-step explanation:
Answer:
For this case we have the following functions transformation:
Vertical expansions:
To graph y = a * f (x)
If a> 1, the graph of y = f (x) is expanded vertically by a factor a.
f1 (x) = (3x) ^ 2
Horizontal translations
Suppose that h> 0
To graph y = f (x-h), move the graph of h units to the right.
f2 (x) = (3x-6) ^ 2
Vertical translations
Suppose that k> 0
To graph y = f (x) + k, move the graph of k units up.
g (x) = (3x-6) ^ 2 + 3
Answer:
expanded horizontally by a factor of 3, horizontal shift rith 6, vertical shift up 3
PLEASE help me answer this question!!
What is 93.195 written in expanded form?
Standard Form:
93.195
Expanded forms can be written like a sentence or stacked for readability as they are here.
Expanded Notation Form:
93.195 =
90
+ 3
+ 0.1
+ 0.09
+ 0.005
Expanded Factors Form:
93.195 =
9 × 10
+ 3 × 1
+ 1 × 0.1
+ 9 × 0.01
+ 5 × 0.001
Expanded Exponential Form:
93.195 =
9 × 101
+ 3 × 100
+ 1 × 10-1
+ 9 × 10-2
+ 5 × 10-3
Word Form:
93.195 =
ninety-three and one hundred ninety-five thousandths
20 points and brainliest!
Express each sentence below as an equation or inequality.
1. Bob has two less than half of the document of points Leroy has.
2. To be in the platinum tier for a ranked video, you must have at least 1000 ranked points.
Answer
Step-by-step explanation:
B=Bob, b=2-(1/2xL)
L= leroy
P<1000
60 POINTS PLS HELP WILL MARK BRAINLIEST EXPLAIN TOO PLS
Step-by-step explanation:
we need to find the slope of each line.
the slope is the ratio y/x indicating how many units y changes, when x changes a certain amount of units.
parallel lines have the same slope.
for a perpendicular line x and y in the slope ratio are put upside down, and the sign is flipped (+ to - and vice versa).
a.
2x - 8y = -24
2x + 24 = 8y
y = 2/8 x + 24/8 = 1/4 x + 3
the slope is 1/4 (always the factor of x).
4x + y = -2
y = -4x - 2
the slope is -4 = -4/1.
this line is therefore perpendicular to the first line.
x - 4y = 4
x - 4 = 4y
y = 1/4 x - 4/4 = 1/4 x - 1
the slope is 1/4.
this line is therefore parallel to the first line.
b.
3x - 9y = 9
9y = 3x - 9
y = 3/9 x - 9/9 = 1/3 x - 1
the slope is 1/3
3y = x + 12
y = 1/3 x + 4
the slope is 1/3.
this line is therefore parallel to the first line.
2x - 6y = 12
6y = 2x - 12
y = 2/6 x - 12/6 = 1/3 x - 2
the slope is 1/3.
this line is therefore parallel to the first line.
Is 3 to the second power + 3 to the 3rd power equal to 3 to the 5th power explain
Answer:
They are not equal.
Step-by-step explanation:
Hi there!
[tex]3^2+3^3[/tex] is not equal to [tex]3^5[/tex]. When two powers with the same base are added together, we cannot add the exponents to get the answer.
We can only add exponents to get the answer when two power with the same base are multiplied.
[tex]3^2+3^3\\= 9+27\\=36[/tex]
[tex]3^5\\=243[/tex]
They are not equal.
I hope this helps!
What is the length of segment HK?
[tex]▪▪▪▪▪▪▪▪▪▪▪▪▪ {\huge\mathfrak{Answer}}▪▪▪▪▪▪▪▪▪▪▪▪▪▪[/tex]
Let's take length of HK = n
By using trigonometry,
[tex] \sin(30) = \dfrac{22}{n} [/tex][tex] \dfrac{1}{2} = \dfrac{22}{n} [/tex][tex]n = 44[/tex]Hence, value of HK = 44 cm
[tex]\\ \rm\Rrightarrow sin\theta=\dfrac{Perpendicular}{Hypotenuse}[/tex]
[tex]\\ \rm\Rrightarrow sin30=\dfrac{JK}{HK}[/tex]
[tex]\\ \rm\Rrightarrow sin30=\dfrac{22}{HK}[/tex]
[tex]\\ \rm\Rrightarrow HK=\dfrac{22}{sin30}[/tex]
[tex]\\ \rm\Rrightarrow HK=44[/tex]
When the same probability is assigned to each outcome of a sample space, the experiment is said to have _______ ________ outcomes.
Answer:
Equally likely
Step-by-step explanation:
Hope this is right!
Sketch the graph of this linear equation
4y = -7x + 1
Answer:
use desmos to graph it
.....
Please help! Geometry practice was lines cut by a transversal. What is the value of X?
Answer:
x = 70
Step-by-step explanation:
1. First, we need to know that the angle with 45 degrees and the angle with (2x-5) degrees are same-side interior angles, meaning they add up to 180 degrees. The equation of this can be written as so: [tex]2x-5+45= 180[/tex]
2. (Solving)
Step 1: Combine like terms.
[tex](2x) +(-5+45) = 180[/tex] [tex]2x + 40 = 180[/tex]Step 2: Subtract 40 from both sides.
[tex]2x + 40 - 40 = 180 - 40[/tex] [tex]2x = 140[/tex]Step 3: Divide both sides by 2.
[tex]\frac{2x}{2} = \frac{140}{2}[/tex] [tex]x = 70[/tex]Step 4: Check if solution is correct.
[tex]2(70) - 5 + 45 = 180[/tex] [tex]140 - 5 + 45 = 180[/tex] [tex]135 + 45 = 180[/tex] [tex]180 = 180[/tex]Therefore, x = 70.
You buy 50 feet of fence to place around a rectangular yard. If the length of the yard is 13 feet, what is the constraint for the width of the yard? Let w represent width *
Step-by-step explanation:
a fence represents the circumference, the perimeter of the lot or yard.
remember, a rectangle has 2 parallel sides called the length, and 2 parallel sides called the width (altogether 4 sides).
therefore
perimeter = 2×length + 2×width
or with short variables
p = 2×l + 2×w
so, now we have 50 ft of fence available for the perimeter. and the length l = 13 ft.
since we have 2 lengths in the perimeter, these 13 ft are twice (2×13 = 26 ft) in the perimeter.
now, how much fence do we have left for the width ?
well, 50 - 26 = 24 ft
and again, this is the size of 2 sides (2×width) :
p = 2×l + 2×w
50 = 2×13 + 2×w = 26 + 2×w
2×w = 24
w = 12 ft
so, the width of the yard must not be bigger than 12 ft.
what do many prefixes and suffixes have in common?
We need to find the combined total of 1,203,000,000 and 2,345,000,000. Write your answer in scientific notation.
PLZ answer quick, 30 points
Answer:
answer in scientific notation.
PLZ answer quick, 30 points
What is the area of the trapezoid?
the area is 160 cm^2
Middle school math pls help
Answer:
2 3/4
Step-by-step explanation:
You need to find the greatest common factor of 4 and 2 then you multiply the 2 by 2 to get a 4 and multiply the 1, then you add those together to get 2 3/4.
Take away -7x from {2y-2x}
Answer:
loading, please wait...
Step-by-step explanation:
Answer:
2y + 5x
Step-by-step explanation:
{ 2y - 2x } - ( - 7x )
= 2y - 2x + 7x
= 2y + 5x
WILL GIVE BRAINLIEST 5x-5=4x+10
Enter the number in scientific notation.
0.00000174 =
X10
Hint: Move the decimal right 6 places.
Answer:
1.74 X 10^-6
Step-by-step explanation:
Hope It helps
Which expression is equivalent to 11+22q+22-5q I’ll give the crown just pls answer
Answer:
[tex]11 + 22q + 22 - 5q \\ = 33 + 17q[/tex]
9+10=21 yay
bababooey
Two similar cups are 3cm and 5cm deep,if the longer cup has 625cm of water.what is the volume of the smaller one
Answer:
81
Step-by-step explanation:
the 5 cm deep cup is 625 which is 5^4
that means that the 3cm deep cup should be 81 which is 3^4
help me with this pelases
Answer:
32 square units
Step-by-step explanation:
You can split this shape into four different shapes:
2 unit by 2 unit triangle
2 unit by 2 unit square
2 unit by 6 unit triangle
To find the length of the big triangle, add the lengths of the small triangle,the square,and the bigger leg of the other triangle:
2 + 2 + 6 = 10
10 unit by 4 unit triangle
Then solve all these areas:
( 2 x 2 ) / 2 = 4 / 2 = 2
2 x 2 = 4
( 2 x 6 ) / 2 = 12 / 2 = 6
( 10 x 4 ) / 2 = 40 / 2 = 20
Then add all these areas together to get the total area of the composite shape:
2 + 4 + 6 + 20 = 32
The GCD of three numbers is 7 and their LCM is 140.If one of the numbers is 20,find the other number
Formula= LCM×GCD of the numbers÷The Given number
Which of the following is a characteristic of a line?
[tex]\huge\rm\purple{\underline{Line:-}}[/tex]
➜ A line is a figure in geometry which has only length with no breadth in a two dimensional plane , and extended infinitely in opposite directions .
➜ A line may be of any length and breadth . An infinite number of combination of long, short , thick or thin lines can Accord to their application unify , divide , balance , or unbalance a pictorial area . This emotion dynamic is set up by lines measures .
⠀⠀━━━━━━━━━━━━━━━━━━━━━━━━━━━━⠀⠀
[tex]\large\sf\orange{\underline{Characteristics\:of\:line:}}[/tex]
Line type : dotted , short dashed , long dashed and continuous line.Line width : extra dark , thick , medium annd thinLine quality : the quality of line depends largely upon the drawing medium⠀⠀━━━━━━━━━━━━━━━━━━━━━━━━━━━━⠀⠀
Please help with this math problem! Will give brainliest! :)
Answer:
9
Step-by-step explanation:
All triangles (3 sided polygons) have angles added up to 180, so we can do the following:
(7x-1) + (7x+8) + (5x+2) = 180° - equation
7x-1 + 7x+8 + 5x+2 = 180° - "distribute"
19x - 1 + 8 + 2 = 180° - combine the x terms
19x + 9 = 180° - combine the integer terms
19x = 171 - subtract 9 form both sides
x = 9 - divide both sides by 19
A company’s stock was selling at $25 a share. A month later, $30 a share. What is the precent increase
Answer:
20 %
Step-by-step explanation:
5 divided by 20 equals x 100
Write the amount in standard scientific notation.
3.65E-2 millimeters
Photo above need help
please help, numbers 14 & 16 only. thank you so much, I’ll mark brainliest.
14. 37/50
16. 13/40
I just converted each fraction into a decimal.
18/25 = 0.72
3/4 = 0.25
11/35 is approximately 0.314
5/14 is approximately 0.357
Then choose a decimal in between and converted it back
What is the rate of change over the interval [-1,2] for f(x)= 4x +1
Answer:
4
Step-by-step explanation:
To find the average rate of change, we have to plug in the difference between the intervals with this equation: f(b) - (fa) / x₂ - x₁
Let f(a) = f(-1) = 4(-1) + 1
f(b) = f(2) = 4(2) + 1
x₂ = 2
x₁ = -1
[4(2) + 1 - (4(-1) + 1)] / [2 - (-1)]
[8 + 1 + 4 - 1] / [3]
12 / 3 = 4