If [tex]a + ar + ar^{2} + ... = 15[/tex] and [tex]a + ar^{2} + ar^{4} + ar^{6} + ... = 9[/tex], then what is the sum of the geometric series [tex]a + ar^{3} + ar^{6} + ar^{9} + ...[/tex]?
Notice that
a + ar³ + ar⁶ + ar⁹ + ... = a + r³ (a + ar³ + ar⁶ + ...)
but the grouped sum on the right side is the same as the sum on the left side. This means that, if the sum we want to compute
a + ar³ + ar⁶ + ar⁹ + ...
converges to S, then
S = a + r³ S
Solve for S :
S - r³ S = a
(1 - r³) S = a
S = a/(1 - r³)
Similarly, knowing the value of the first sum, we have
a + ar + ar² + ar³ + ... = a + r (a + ar + ar² + ...)
but we know
a + ar + ar² + ... = 15
so
15 = a + 15r
and knowing the value of the second sum, we have
a + ar² + ar⁴ + ... = a + r² (a + ar² + ar⁴ + ...)
so
9 = a + 9r²
Solve these two equations for a and r :
15 = a + 15r ⇒ a = 15 - 15r
9 = a + 9r² ⇒ 9 = 15 - 15r + 9r² ⇒ 3r² - 5r + 2 = (3r - 2) (r - 1) = 0
⇒ r = 2/3 or r = 1
We cannot have r = 1, since that would mean
a + a + a + ... = 15
since the left side is adding infinitely many copies of some constant; such a sum can never converge to a finite number. So it must be that r = 2/3, and it follows that a = 15 - 15 (2/3) = 5.
Then the sum we want has a value of
S = 5/(1 - (2/3))³ = 135/19
Answer:
Step-by-step explanation:
a + ar¹ + ar² + ar³ + ... = 15 #1
r(a + ar¹ + ar² + ar³ + ...) = 15r
ar¹ + ar² + ar³ + ... = 15r #2
subtract #2 from #1
a = 15 - 15r
a + ar² + ar⁴ + ar⁶ + ... = 9 #3
r²(a + ar² + ar⁴ + ar⁶ + ... )= 9r²
ar² + ar⁴ + ar⁶ + ... = 9r² #4
subtract #4 from #3
a = 9 - 9r²
15 - 15r = 9 - 9r²
9r² - 15r + 6 = 0
3r² - 5r + 2 = 0
r = (5 ±√(5² - 4(3)(2))) / (2(3)
r = (5 ± 1) / 6
r = 1
or
r = 4/6 = 2/3
if r = 1 then a = 15 - 15(1) = 0 and a series of zeros added together can never equal 15 so r = 2/3 must be true.
a = 15 - 15(2/3) = 5
a + ar³ + ar⁶ + ar⁹ + ... = C #5
r³(a + ar³ + ar⁶ + ar⁹ + ...) = Cr³
ar³ + ar⁶ + ar⁹ + ... = Cr³ #6
subtract #6 from #5
a = C - Cr³
a = C(1 - r³)
C = a/(1 - r³)
C = 5/(1 - (2/3)³)
C = 5/(1 - 8/27)
C = 5/(19/27)
C = 5(27/19)
C = 135/19 = 7.1052631578947368....
End of semester geometry Semester A question 6
Select the correct answer.
Given:
Prove:
By the linear pair theorem, is supplementary to . Since , by the definition of congruence, . Applying the substitution property of equality, . Simplifying the equation, .
What step is missing from this proof?
A.
is supplementary to by the transitive property.
B.
by the definition of supplementary angles.
C.
by the definition of congruence.
D.
by the linear pair theorem.
Answer:
by the definition of congruence.
Step-by-step explanation:
for plato
2. Calculate an expression for dy/dx and d2y/dx2 in terms of t if the parametric pair is given as tan(x) = e^at and e^y = 1 + e^2at
Hint: Turn tan(x) = e^at and e^y = 1 + e^2at into expressions suited to your calculation.
I assume a is a constant. If tan(x) = exp(at) (where exp(x) means eˣ), then differentiating both sides with respect to t gives
sec²(x) dx/dt = a exp(at)
Recall that
sec²(x) = 1 + tan²(x)
Then we have
(1 + tan²(x)) dx/dt = a exp(at)
(1 + exp(2at)) dx/dt = a exp(at)
dx/dt = a exp(at) / (1 + exp(2at))
If exp(y) = 1 + exp(2at), then differentiating with respect to t yields
exp(y) dy/dt = 2a exp(2at)
(1 + exp(2at)) dy/dt = 2a exp(2at)
dy/dt = 2a exp(2at) / (1 + exp(2at))
By the chain rule,
dy/dx = dy/dt • dt/dx = (dy/dt) / (dx/dt)
Then the first derivative is
dy/dx = (2a exp(2at) / (1 + exp(2at))) / (a exp(at) / (1 + exp(2at))
dy/dx = (2a exp(2at)) / (a exp(at))
dy/dx = 2 exp(at)
Since dy/dx is a function of t, if we differentiate dy/dx with respect to x, we have to use the chain rule again. Suppose we write
dy/dx = f(t)
By the chain rule, the derivative is
d²y/dx² = df/dx
d²y/dx² = df/dt • dt/dx
d²y/dx² = (df/dt) / (dx/dt)
d²y/dx² = 2a exp(at) / (a exp(at) / (1 + exp(2at)))
d²y/dx² = 2 (1 + exp(2at))
Round 1276 to the nearest ten?
Answer:
1280
Step-by-step explanation:
look at the 6 in the number
is it 4 or less
or is it 5 or more
as the number is 6 ( more than 5)
then you are going to add 1 to the tens
so the final answer is 1280
I really hope this helps and have a great day
What is the answer to the following question?
Answer:
SAS similarity
Step-by-step explanation:
You have two side ratios and the angle between the sides with those ratios. The appropriate similarity theorem is SAS.
Which number line represents the number 0.7
Answer:
What number like?
Answer: The answer is C.
The question I assume you are talking about is in the image.
Complete the table of ordered pairs for the linear equation
Answer:
Step-by-step explanation:
See attachment.
John finished his English assignment in 1/3 hours. Then he completed his chemistry assignment in 2/5 hours. What was the total amount of time John spent doing these two assignments?
these two assignments?
Answer: 11/15 hours
Step-by-step explanation:
Find LCD (lowest common denominator)
1/3=5/15
2/5=6/15
6/15+5/15=11/15 hours or 44 minutes
100 POINTS WILL AWARD BRAINLIEST PLEASE HELP
(IMAGE INCLUDED)
In the image below, the m
Answer: 67°.
Step-by-step explanation:
1) m∠PMR=m∠LMN=3x+19°;
2) m∠LMN+m∠LMP=180°, it can be written as 3x+19+9x-31=180;
3) if to solve the equation 3x+19+9x-31=180, then x=16;
4) m∠PMR=3x+19=48+19=67°.
Find x
Linear pairs are supplementary
9x-31+3x+19=18012x-12=180x-1=15x=15+1=16So
opposite angles are equal
m=9(16)-31=144-31=113°Which statement is the correct interpretation of the inequality −4 > −5? (5 points)
Question 7 options:
1)
On a number line, −4 is located to the left of 0 and −5 is located to the right of 0.
2)
On a number line, −4 is located to the right of 0 and −5 is located to the left of 0.
3)
On a number line, −4 is located to the right of −5.
4)
On a number line, −4 is located to the left of −5.
Answer:
On a number line, -4 is located to the right of -5
Step-by-step explanation:
Answer:
On a number line, -4 is located to the right of -5.
Step-by-step explanation:
this is correct, also I don't take credit for it the person above me is the original. :)
One-celled organisms that are more complex than bacteria are called viruses.
Question 6 options:
True
False
Answer:
false
Step-by-step explanation:
bacteria are bigger and more complex than viruses.
The polygons below are similar. Find the value of y
Answer:
y = 21
Step-by-step explanation:
what is the perimeter of a rectangular whose area is 92 square mm and length is 11.5 mm
We are given –
Area of rectangular is = 92 square mmLength of rectangular is = 11.5 mmWe are asked to find out the value of perimeter of the rectangular! Let's find it's breadth first!We know that–
[tex]\qquad[/tex][tex] \purple{\bf\longrightarrow Area_{(Rectangular)} = Length \times Breadth} [/tex]
[tex]\qquad[/tex][tex] \sf \longrightarrow 92 = 11.5 \times Breadth [/tex]
[tex]\qquad[/tex][tex] \sf \longrightarrow Breadth = \cancel{\dfrac{92}{11.5}}[/tex]
[tex]\qquad[/tex][tex] \purple{\bf \longrightarrow Breadth = 8 \: mm}[/tex]
Now, Perimeter of rectangular –
[tex]\qquad[/tex][tex] \pink{\bf \longrightarrow Perimeter _{(Rectangular) } = 2( Length + Breadth) }[/tex]
[tex]\qquad[/tex][tex] \sf \longrightarrow Perimeter _{(Rectangular) }= 2( 11.5 +8) [/tex]
[tex]\qquad[/tex][tex] \sf \longrightarrow Perimeter _{(Rectangular) } = 2 \times 19.5[/tex]
[tex]\qquad[/tex][tex]\pink{ \bf \longrightarrow Perimeter _{(Rectangular) } = 39\: mm}[/tex]
Henceforth, Perimeter of rectangular is 39mm._______________________________________
Answer:
Perimeter = 39 mmStep-by-step explanation:
In this question we are given with ,
Area of rectangle = 92 sq mmLength of rectangle = 11.5 mmAnd we're asked to find the perimeter of rectangle .
Solution : -
We know that ,
[tex] \pink{ \boxed{\sf{Area \: of \: rectangle = L × W}}}[/tex]
Where ,
L = length of rectangleW = width of rectangleSo from this formula we can find the width of rectangle and equating it with 92 ( because 92 is the given area ) . Substituting value of length :
[tex] \longmapsto \: 11.5 \times W = 92[/tex]
Step 1 : Multiplying 11.5 and W :
[tex] \longmapsto \: 11.5W = 92[/tex]
Step 2 : Dividing by 11.5 on both side :
[tex] \longmapsto \: \frac{ \cancel{11.5}W}{ \cancel{ 11.5}} = \frac{92}{11.5} [/tex]
On further calculations we get :
[tex] \longmapsto \: W = \cancel{\frac{920}{115} }[/tex]
Step 3 : By cancelling 920 by 115 we get :
[tex] \longmapsto \: \purple{\boxed{ \bold{W = 8 \: mm}}}[/tex]
Therefore , width of rectangle is 8 mm .Finding Perimeter :We know that ,
[tex] \blue{ \boxed{ \sf{Perimeter \: of \: rectangle = 2 ( L +W )}}}[/tex]
Where ,
L = Length of rectangle W = Width of rectangleSubstituting values in formula :
[tex] \longrightarrow \: 2(11.5 + 8)[/tex]
Step 1 : Solving parenthesis :
[tex] \longrightarrow \: 2(19.5)[/tex]
Step 2 : Multiplying 2 with 19.5 :
[tex] \longrightarrow \: \green{\boxed{ \bold{39 \: mm}}}[/tex]
Therefore, perimeter of rectangle is 39 mm. #Keep LearningSally used a ruler to draw the segment ab she then opened her compass to the length of ab and drew a circle keeping thw compass open to the same length she marked off equal parts along the circle as shown in the following figure using some or all of the points of intersection and point a which of the following regular polygons could she construct
Answer:
Step-by-step explanation:
Equilateral Triangle
Regular Hexagon
PLEASE HELP THIS WILL GIVE 50 POINT AND BRAINLIEST！！！！！!
In introductory physics, a typical cavendish balance for measuring the gravitational constant G use metal masses 2.39kg and 16g whose center are separated by 6.81cm calculate the gravitation force between these forces, treating each as a point mass located at the center of the sphere.
gravitational constant =6.67259 × 10⁻¹¹N·m²/kg²
Answer:
the asnswer is -67463 to the power of 4
Step-by-step explanation:
Pls help with these questions!
Answer:
I don't know sorry
Which number comes next in the series? 7, 10, 10, 13, 16, 16,
Answer:
19
Step-by-step explanation:
In the first number, they add three and that number is repeated.
To further explain, the series would continue like this:
7, 10, 10, 13, 16, 16, 19, 22, 22...
Hope this helps! Brainliest would be appreciated!
Grandma Maude wanted a tall cake for her 80th birthday. The baker made a cake with 4x3 – x layers. Each layer was 2 + 3x2 cm. Which expression shows the total height of Grandma Maude’s cake?
Using proportions, it is found that the expression that shows the total height of Grandma Maude’s cake is:
[tex]t = 12x^5 + 5x^3 - 2x[/tex]
This question is solved by proportions, using a rule of three.The measure of one layer, in cm, was of [tex]2 + 3x^2[/tex]. In total, there were [tex]4x^3 - x[/tex] layers.The rule of three is:
1 layer - [tex]2 + 3x^2[/tex] cm
[tex]4x^3 - x[/tex] layers - t cm
Applying cross multiplication:
[tex]t = (4x^3 - x)(2 + 3x^2) = 12x^5 + 5x^3 - 2x[/tex]
To learn more about proportions, you can take a look at https://brainly.com/question/24372153
Answer:
12x5 + 5x3 – 2x cm
Step-by-step explanation:
A 20 cm long thin wire is used to form a square. What is the area of the square?
Answer:
25cm^2
Step-by-step explanation:
A square has 4 equal sides so each side will be 20/4 = 5cm
Easily calculate the area by taking 5 x 5 = 25cm^2
what is the equation for: 32 is the sum of a number x and 9
Answer:
x + 9 = 32
Step-by-step explanation:
answer for x is 32 - 9 = 23
Planet Q Is 3.6 x 10^6km
Planet R Is 1.6 x 10^7km
What Is The Shortest Distance Between Planet Q And R (Answer In Standard Form)
I've Been Stuck On This Question For A Vast Amount Of Time.
Answer:
1.64×10⁷ km
Step-by-step explanation:
We know this is a triangle rectangle, so we apply the Pythagorean Theorem
QR=sqrt(PR²+PQ²)
Solving it, we will encounter in
sqrt(2.56×10¹⁴+1.296×10¹³)
to solve it, we need to bring the two numbers at the same power of ten, so we move 1.296 of one place to the left:
sqrt(2.56×10¹⁴+0.1296×10¹⁴)
we can now add the two numbers:
sqrt(2.6896×10¹⁴)
to solve this, we do the square root of 2.6896 and divide 14 by two (the index of the root is 2, so for roots' properties we can divide 14 by 2 - the index)
which gives us
1.64×10⁷
Which property explains why these two expressions are equal? 7 + q + 4 = 4 + 7 + q
Simplify the expression. Write your answer as an improper fraction.
-2/3x divided by (-4/9)
Answer:
3/2x
Step-by-step explanation:
−
2
3
x
−
4
9
=
3
2
x
To make cupcakes Leah uses 1/2 cup of chocolate chips for evry 1 1/3 cups of batter if Leah uses 3/4 cups of chocolate chips how many cups of batter should she use
Answer:
2
Step-by-step explanation:
Leah uses 1/4 cup of chocolate chips for 2/3 cups of batter, so if Leah uses 3/4 cup of chocolate chips, she should use 2 cups of batter.
Write a word phrase to represent the numerical expression below.
(7-3) ÷ 2
Answer:
the quantity seven minus three divided by two
Step-by-step explanation:
its math
someone please help me giving brainlist
Answer:
A
Step-by-step explanation:
atteo spends a total of 38 min exercising. He walks for 6 min to warm up and then runs at a constant rate of 8 min per mile for the rest of the time. Matteo says that he ran 4.75 mi. Is he correct? Explain your reasoning.
Answer:
he is incorrect.
no one can run for 5 miles everyday.
if so, my man atteo is a legend.
A group of pigs and ducks has a total of 40 feet. There are twice as many ducks as pigs. How many of each animal are there?
The question relates to the total number of pairs of feet ducks and a pigs
are known to have.
There are 5 pigs and 10 ducksReasons:
The length of the group of ducks and pigs = 40 feet
Number of ducks = Twice the number of pigs
Each duck has 2 feet.Each pig has 4 feet.Let x represent the number of pigs, and let y represent the number of ducks, we have;
y = 2 × x
4·x + 2·y = 40
Which gives;
4·x + 2 × (2·x) = 40
8·x = 40
x = 40 ÷ 8 = 5
The number of pigs, x = 5y = 2 × x
Therefore;
y = 2 × 5 = 10
The number of ducks, y = 10Learn more about solving word problems in mathematics here:
https://brainly.com/question/25272317
A house costs 36% more today than when it was built.If the cost of the house today is $333,200, find the cost when it was built.
Answer:
$245,000
Step-by-step explanation:
Let the intial cost = x
[tex]100\% + 36\% = 136\% = 1.36[/tex]
[tex]1.36x = 333200[/tex]
[tex]x = \frac{333200}{1.36} [/tex]
[tex]x = 245000[/tex]
What is the value of the expression 2z ÷ [z – (z – 4)] when z = 8?
2z ÷ [z – (z – 4)]
Sub z=8 into the expression
2(8) ÷ [8 – (8 – 4)] = 16 ÷ 4
= 4