Let
[tex]S_n = \displaystyle \sum_{k=0}^n r^k = 1 + r + r^2 + \cdots + r^n[/tex]
where we assume |r| < 1. Multiplying on both sides by r gives
[tex]r S_n = \displaystyle \sum_{k=0}^n r^{k+1} = r + r^2 + r^3 + \cdots + r^{n+1}[/tex]
and subtracting this from [tex]S_n[/tex] gives
[tex](1 - r) S_n = 1 - r^{n+1} \implies S_n = \dfrac{1 - r^{n+1}}{1 - r}[/tex]
As n → ∞, the exponential term will converge to 0, and the partial sums [tex]S_n[/tex] will converge to
[tex]\displaystyle \lim_{n\to\infty} S_n = \dfrac1{1-r}[/tex]
Now, we're given
[tex]a + ar + ar^2 + \cdots = 15 \implies 1 + r + r^2 + \cdots = \dfrac{15}a[/tex]
[tex]a^2 + a^2r^2 + a^2r^4 + \cdots = 150 \implies 1 + r^2 + r^4 + \cdots = \dfrac{150}{a^2}[/tex]
We must have |r| < 1 since both sums converge, so
[tex]\dfrac{15}a = \dfrac1{1-r}[/tex]
[tex]\dfrac{150}{a^2} = \dfrac1{1-r^2}[/tex]
Solving for r by substitution, we have
[tex]\dfrac{15}a = \dfrac1{1-r} \implies a = 15(1-r)[/tex]
[tex]\dfrac{150}{225(1-r)^2} = \dfrac1{1-r^2}[/tex]
Recalling the difference of squares identity, we have
[tex]\dfrac2{3(1-r)^2} = \dfrac1{(1-r)(1+r)}[/tex]
We've already confirmed r ≠ 1, so we can simplify this to
[tex]\dfrac2{3(1-r)} = \dfrac1{1+r} \implies \dfrac{1-r}{1+r} = \dfrac23 \implies r = \dfrac15[/tex]
It follows that
[tex]\dfrac a{1-r} = \dfrac a{1-\frac15} = 15 \implies a = 12[/tex]
and so the sum we want is
[tex]ar^3 + ar^4 + ar^6 + \cdots = 15 - a - ar - ar^2 = \boxed{\dfrac3{25}}[/tex]
which doesn't appear to be either of the given answer choices. Are you sure there isn't a typo somewhere?
This graph shows a proportional relational between the distance traveled by Tim on a road trip and the number
of hours for which he traveled.
Distance Traveled by Tim
250
200
150
Distance (mi)
100
50
0
10
6 8
Number of Hours
Which statement identifies the correct slope and the correct interpretation of the slope for this situation?
50
A The slope of the line is
1
so the distance traveled by Tim is miles every hour.
50
B The slope of the line is so the distance traveled by Tim is miles every hours.
1
1
The slope of the line is
50
so the distance traveled by Tim is miles every hour.
Using the proportional relationship, it is found that the correct statement regarding the slope of the relation is:
The slope of the line is of 25, so the distance traveled by Tim is 25 miles every hour.
What is a proportional relationship?A proportional relationship is a function in which the output variable is given by the input variable multiplied by a constant of proportionality, that is:
y = kx
In which k is the constant of proportionality, which is also called the slope.
Researching the problem on the internet, it is found that in 2 hours, Tim traveled 50 miles, hence:
k = 50/2 = 25.
The slope of the line is of 25, so the distance traveled by Tim is 25 miles every hour.
More can be learned about proportional relationships at https://brainly.com/question/10424180
Oscar enjoys many different sports. Today, Oscar jogged for 6 minutes, rode his bike for 23 minutes, and played tennis for 49 minutes.
How much time did Oscar spend playing sports today?
Answer: 78
Step-by-step explanation:
just add the time all together then this will be your answer.
solve for x ~
[tex]81x - 27x = 9[/tex]
thankyou ~
Listen ducky: 81x-27x=9
54x=9
X=1
---
6
[tex]\qquad\qquad\huge\underline{{\sf Answer}}[/tex]
Here we go ~
[tex]\qquad \tt \dashrightarrow \:81x - 27x = 9[/tex]
[tex]\qquad \tt \dashrightarrow \:9(9x - 3x )= 9[/tex]
[tex]\qquad \tt \dashrightarrow \:(9x - 3x )= 1[/tex]
[tex]\qquad \tt \dashrightarrow \:6x = 1[/tex]
[tex]\qquad \tt \dashrightarrow \:x = \cfrac{1}{6} [/tex]
A solid has volume 7 cubic units. The equation k =
represents the scale factor of k by which
the solid must be dilated to obtain an image with volume V cubic units. Select all points which are
on the graph representing this equation.
a. (0,0)
e. (14,2)
b. (1,1)
f. (49,2)
C. (1,7)
(
g. (56,2)
(71)
h (27,3)
The volume of the solid is the amount of space in it.
The points on the graph are (0,0) and (1,7)
How to determine the points?The volume is given as:
V1 = 7
The scale factor is given as:
k
Let the new volume be V.
So, we have:
V = V1 * k^3
Substitute 7 for V1
V = 7k^3
Divide both sides by 7
k^3 = V/7
Take the cube root of both sides
k = (V/7)^1/3
Next, we test the options
a. (0,0)
k = (0/7)^1/3
k = 0 --- this is true
b. (1,1)
k = (1/7)^1/3
k = 0.05 --- this is false
c. (1,7)
k = (7/7)^1/3
k = 1 --- this is true
d (7, 1)
k = (1/7)^1/3
k = 0.05 --- this is false
e. (14,2)
k = (2/7)^1/3
k = 0.10 --- this is false
f. (49,2)
k = (2/7)^1/3
k = 0.10 --- this is false
g. (56,2)
k = (2/7)^1/3
k = 0.10 --- this is false
h (27,3)
k = (3/7)^1/3
k = 0.14 --- this is false
Hence, the points on the graph are (0,0) and (1,7)
Read more about volumes at:
https://brainly.com/question/12232378
Read the sentence.
When my cousin Jason suddenly showed up out of the blue, it was an unexpected surprise because we had no idea that he was coming to the special event that our family had planned ahead of time.
Which revision most effectively shortens the sentence to make it less repetitive and more precise?
Drag the revision into the box.
The first sentence is correct.
Answer: We were surprised when my cousin jason attended our family reunion.
Step-by-step explanation: I took the k12 quiz and got it right.
I don’t get 2 and 3 I really need help
Answer:
Step-by-step explanation:
Opposite angles of a cyclic quadrilateral sum to 180 so
m < x = 180-105 = 75 degrees
m < y = 180 - 95 = 85 degrees.
Select the correct answer.
A company designs a new water pitcher. They take a picture of the pitcher for their promotional materials. What is the approximate area filled
by the water in the picture on the flyers?
5 in
4 in
3 in
12 in
8 in
О А.
about 48 in?
B. about 66 in?
Oc about 32 in?
OD. about 80 in?
Applying the area of trapezium, the approximate area filled by water in the picture would be close to: about 48 in.²
What is the Area of a Trapezium?Area of trapezium = 1/2(a + b)h.
The following are dimensions of the area of the trapezium filled with water:
Level of water (a) = median of the bigger trapezium = (3 + 8)/2 = 5.5 = 6 in.b = 8 in.Height ≈ 12/2 = 6 in.Area = 1/2(6 + 8) × 6 = 42 in.²
Therefore, applying the area of trapezium, the approximate area filled by water in the picture would be close to: about 48 in.²
Learn more about area of trapezium on:
https://brainly.com/question/16904048
Find the product of 0.000879 and 1.028
Answer:
0.008791747
Step-by-step explanation:
0.000879
x 1.028
-------------------
0167
01580
000000
08790000
---------------------
8791747
0.000879 and 1.028 have NINE decimal places total, so add NINE decimal places from the LEFT to the above answer.
Final answer: 0.008791747
I hope this helps!
The area of a circle is 9π m². What is the circumference, in meters? Express answer in terms of π
Answer:
20.25 pi
Step-by-step explanation:
The circumference of a circle is given by
C = 2 * pi *r
9 pi = 2 * pi *r
Divide each side by 2pi
9pi / 2 pi = 2 pi r / 2 pi
4.5 = r
Now we can find area
A = pi r^2
= pi ( 4.5)^2
20.25 pi
A coordinate plane with 3 points plotted at (10, 18), (14, 24), and (18, 30).
Which statement describes the function through the points?
Answer:
They are all even
Step-by-step explanation:
ARGENTTTT!!!!
Which statement describes the difference in the distance
travelled by the two cars when 15 gallons of gasoline is used?
Both cars will have travelled the same distance.
Answer:
D. A travels 90 miles more
Step-by-step explanation:
The graph tells you Car A travels 140 miles on 5 gallons of gas. (There is a point marked there on the graph.)
The table tells you Car B travels 99 miles on 4.5 gallons of gas. When that is multiplied by 5/4.5 = 10/9 to get the equivalent for 5 gallons, we see that Car B could travel 10/9 × 99 = 110 miles on 5 gallons of gas.
__
Already on 5 gallons of gas, Car A has traveled 30 miles farther. The only sensible answer choice is ...
D. Car A travelled 90 miles more than Car B [when 15 gallons are used].
What is the mode of the data represented in this line plot?
Enter your answer in the box.
Answer:
3
Step-by-step explanation:
The mode is 3 because it is the most commonly repeating number in the data set.
The mode of the data represented in this line plot is 3
How to determine the mode?From the line plot, we have the following frequency table
Data Frequency
1 2
2 3
3 4
4 3
In the above frequency table, the entry 3 has the highest frequency of 4
Hence. the mode of the data is 3
Read more about mode at:
https://brainly.com/question/14532771
#SPJ2
how i do this? help pls
For questions 4 – 6, name all of the radii of the circle shown in the figure.
For questions 7 – 10, find the measure of the radius of the circle. 7. 8.
For questions 4 – 6, name all of the radii of the circle shown in the figure.
For questions 7 – 10, find the measure of the radius of the circle. 7. 8.
Answer:
center(-2,5) r=6
Which expression is equivalent?
Answer:
The Third one.
Step-by-step explanation:
So, Turn the mixed number, 2 1/4, to an improper fraction, 9/4. Then match the answer!
If you need anything else, just message me!
Have a nice day! :)
Please help! First answer correct BRAINLIEST
Answer:
8 in
Step-by-step explanation:
Area=(1/2)*(p1+p2)*d
Susan gave 1/6 of a pizza to her brother. She gave 1/3 of what was left to
her mother. Susan ate the rest herself. What fraction of the pizza did Susan
eat?
°1/2
°1/9
°1/18
°5/9
Answer:1/2
Step-by-step explanation: after giving some to her brother she has 5/6 left to take 1/3 from that you need to make it have a common denominator. to do that multiply it by 2 so you get 2/6. then do 5/6-2/6=3/6. 3/6 simplied is 1/2.
Let g: R+R+ be defined by g(x) = x2 +1. Determine the domain, co-domain and range of g.
Domain:-
For any real x function is not undefined .
Domain belongs to RCodomain belongs to R+
Range:-
x² always ≥0Hence
range=[1,oo)In a Right Triangle, if the measurement for the base (B) equals a negative 6 and the hypotenuse equals (C) equals 12, then what is the value of the side (A)?
A swimming pool is in the shape of a cylinder 7 feet deep and 9 feet in radius. Find the exact volume of the pool.
Answer:
16
Step-by-step explanation:
Hope this helps
Write a proportion for the statement.
$118 per week is proportional to $472 per 4 weeks. The proportion is _____?
A proportion is a mathematical comparison between two ratios.
A ratio is a comparison between two numbers.
For instance, the ratio of apples to bananas in a fruit basket is 1 apple to 2 bananas:
[tex]\dfrac{1\hspace{3}apple}{2\hspace{3}bananas}[/tex]This is proportional to the ratio of apples to bananas in another fruit basket, which is 2 apples to 4 bananas:
[tex]\dfrac{2\hspace{3}apples}{4\hspace{3}bananas}[/tex]To write a proportion from this information, we can compare the ratios by makign them equal to each other:
[tex]\dfrac{1\hspace{3}apple}{2\hspace{3}bananas}=\dfrac{2\hspace{3}apples}{4\hspace{3}bananas}[/tex][tex]\dfrac{1}{2}=\dfrac{2}{4}[/tex]Solving the QuestionWe're given:
$118 per week$472 per 4 weeksThese are proportional$118 per week is one ratio:
[tex]\dfrac{118\hspace{3}dollars}{1\hspace{3}week}[/tex]$472 per 4 weeks is another ratio:
[tex]\dfrac{472\hspace{3}dollars}{4\hspace{3}weeks}[/tex]Now, set them equal to each other to create a proportion:
[tex]\dfrac{118\hspace{3}dollars}{1\hspace{3}week}=\dfrac{472\hspace{3}dollars}{4\hspace{3}weeks}\\\\\dfrac{118}{1}=\dfrac{472}{4}[/tex]Answer:Therefore, the proportion is [tex]\dfrac{118}{1}=\dfrac{472}{4}[/tex].
HELP ASAP FIRST GET BRAINIEST
!!!!!!!!!!!!!!!!!!!!!!!!!
Answer:
C
Step-by-step explanation:
just imagine that they are folding and what they would fold into.
this kind of question is a memory test.
Hope this helps!
Brainliest please!
Please fully rate!
Have a great day!
PLEASE HELP PLEASE HELPP
Answer:
B
Step-by-step explanation:
Coordinates are (x,y) so just look at the table on how the x-value compares.
Answer:
answer is B
pls give me branlist
X - 8 = 1 what does X equal to?
Answer:
9
Step-by-step explanation:
To answer this we add 8 to both sides:
x-8+8 = 1+8
x = 9
This is our answer
Given:-
X - 8 = 1To find :-
The value of xSolution :-
X - 8 = 1
Transferring 8 to Right sideX = 1 + 8 ( signs change when transferred to opposite side )
X = 9
What is the slope of a line perpendicular to the line whose equation is
3x – 3y = 63. Fully simplify your answer.
4 + (x + y) = (4 + x) + y =
what are the properties?
Answer:
Distributive
Step-by-step explanation:
can someone help me solve this?
Answer:
[tex]\dfrac{7\pi}{18}[/tex]
Step by step explanation:
[tex]70^{\circ}=\dfrac{70\pi}{180}~~\text{radians}=\dfrac{7\pi}{18}~~\text{radians}[/tex]
can someone solve this
Answer:
check the image for solution
Step-by-step explanation:
I need help please I am good at read not math so please help I don't get it ):
Answer: 782
Step-by-step explanation
It says it made a w
Answer:
Part A: Andy is incorrect
Part B: 3/20
Step-by-step explanation:
Part A: Since Andy:
Actions dvds = 3/5
Comedy dvds = 1/4
or = Add
3/5 + 1/4
12 + 5 /20 = 17/20
Thus, it doesn't equal to 4/9.
Part B: He needed to make 3/5 = 12/20 and 1/4 = 5/20. Adding these together would be 17/20, making 3/20 of them NOT action or comedy. 17/20 + 3/20 = 20/20 or the total.
Therefore, 3/20 of them Not action or comedy.
~lenvy~
What is the volume of the prism?
Answer:
[tex]the \: volume \: of \: a \: rectangular \: prism \: is \\ l \times w \times h \\ in \: this \: graph \: \\ l = 2 \frac{1}{4} \\ w = 2 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \ \: \: \: \: \: h = 4 \\ 2 \frac{1}{4} \times 2 \times 4 \\ 4 \frac{1}{2} \times 4 = 18 \\ the \: volume \: is \: 18 {in}^{3} [/tex]