The given series is a geometric series. The sum of its first 6 terms is 728.
To determine whether the series is an arithmetic or geometric one, we need to check how the consecutive terms relate to each other.
If it is an arithmetic series, then:
a(n) = a(n-1) + d
where d is the common difference.
If it is a geometric series, then:
a(n) = a(n-1) . r
where r is the common ratio.
The given series is:
2+6+18+ . . . .
and the number of terms is 6.
In the given series:
a(1) = 2
a(2) = 6
a(3) = 18
Notice that
a(2)/a(1) = 6/2 = 3
a(3)/a(2) = 18/6 = 3
Since the ratio of consecutive terms are constant, then the given series is a geometric series. Furthermore, its ratio, r = 3
Sum formula for the first n term of a geometric series is:
S(n) = a(1) . (rⁿ -1)/(r -1)
Substitute n = 6, a(1) = 2, and r = 3
S(6) = 2 . (3⁶ - 1)/(3 - 1)
S(6) = 2. 364
= 728
We can also evaluate the sum directly. Write all the terms until n = 6.
a(4) = 18 . 3 = 54
a(5) = 54 . 3 = 162
a(6) = 162 . 3 = 486
Hence, the series is:
2+6+18+54+162+486 = 728
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describe the transformation then graph f(x)=x+4; g(x)=f(x-2)
Answer:
Use This it will help
Step-by-step explanation:
Determine the values of y in the equation y2 = 121.
pls help
Answer:
y = 11
Step-by-step explanation:
Let's solve your equation step-by-step.
[tex] \rightarrow \sf \: y {}^{2} =121[/tex]
Step 1: Take square root.
[tex] \sf \: y=± \sqrt{121} [/tex]
y=11 or y=−11
Which of the following is equivalent to (10x-3)-(6x-2)?
.4x-1
.4x-5
.16x-1
.16x-5
In a direct variation, y=12when x=3.Write a direct variation equation that shows the relationship between x and y
The direct variation equation between x and y is 4x = y.
Here, we are given that x and y are in direct variation. In other words, this means that x and y are directly proportional to each other.
Thus, if k is a constant term, then the relation between x and y can be represented as-
x = ky
Now, we are given that if y = 12, x = 3
⇒ 3 = 12k
12k = 3
k= 3/12
k= 1/4
Thus, we found the value of k, the relation will become-
x = y/4
or 4x = y
Thus, the direct variation equation between x and y is 4x = y.
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Find the 32 nd term of each sequence. 9,4,-1,-6,-11, ...........
The 32th term of the arithmetic sequence AP is found to be -146.
What is the definition of arithmetic progression?An arithmetic progression (AP) is a succession in which each successive term has the same distinctions.
Each term in an arithmetic progression (except the first) is derived by adding a certain number to a term before the first one.The following arithmetic progression formulas are frequently used to solve various AP problems for the initial term 'a' of an AP along with the common difference 'd.'The common difference of AP is 'd': d = a2 - a1 = a3 - a2 = a4 - a3 =...... = a - an-1. nth term of an AP: an = a + (n - 1) dThe sum of an AP's n terms is Sn = n/2(2a+(n-1)d) = n/2(a + l), where l is the final term of the AP.Now, to answer the question;
The given sequence of the numbers is-
9,4,-1,-6,-11, ...........
There are total 32 terms in the series for which we have to find 32nd number.
Let the first term be 'a₁' = 9.
Let the second term be 'a₂' = 4
Let the third term be 'a₃' = -1.
The common difference of between the two consecutive terms is equal of an AP is 'd'.
d = a₂ - a₁
Substitute the values in the above equation;
d = 4 - 9
d = -5
Now, the 32th term is calculated by the nth term formula;
nth term of an AP: an = a + (n - 1) dWhere, total number of terms is; n = 32First term is a = 9common difference d = -5Now, substitute all the value in the formula,
a₃₂ = a + (n - 1) d
a₃₂ = 9 + (32 - 1)(-5)
a₃₂ = 9 - 160 + 5
a₃₂ = -146.
Therefore, the 32nd term of the arithmetic sequence is estimated as -146.
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What is the distance between 4/5 and -1/6 on a number line?
The distance between the point [tex]\frac{4}{5}[/tex] and [tex]\frac{-1}{6}[/tex] is [tex]\frac{9}{10}[/tex] or 0.9.
Given point are [tex]\frac{4}{5}[/tex] and [tex]\frac{-1}{6}[/tex]
In decimal form,
[tex]\frac{4}{5}=0.8[/tex]
[tex]\frac{-1}{6} =-0.1[/tex]
Plot that two point on a number line.
Number line means a straight line with numbers arranged at equal segments or intervals throughout its length is known as a number line. A number line is often shown horizontally and can be extended indefinitely in any direction.
In the figure we can see the number plotting.
The points between those points is the distance between the [tex]0.8[/tex] and [tex]-0.1[/tex].
Therefore, the distance between the point [tex]\frac{4}{5}[/tex] and [tex]\frac{-1}{6}[/tex] is [tex]\frac{9}{10}[/tex] or 0.9.
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Please help!! Part 2 Use the vertical line test to determine if each graph shows a relation that is a function.
Answer:
first graph = NOT a functionsecond graph = is a function===============================================
Reason:
In the first graph, it is possible to draw a single vertical line through more than one point on the blue graph. For instance, draw a vertical line through 1 on the x axis.
Because this happens, we consider the graph to "fail the vertical line test". This leads to the graph not being a function.
A function is only possible if any input in the domain leads to exactly one and only one output.
The input x = 1 leads to multiple outputs y = 3 and y = -3 simultaneously.
---------------------------
On the other hand, the second graph doesn't have this issue happening. It's impossible to draw a single vertical line through more than one point on the blue graph.
Therefore, this graph passes the vertical line test and it is a function. Whatever x input you pick, it leads to one y output only.
Multiply. (−11314)⋅(159) What is the product? Enter your answer in the box.
Answer:
the answer is -3
Step-by-step explanation:
I took the quiz on k12
Write a two-column proof
Given: ∠P ≅ ∠T, ∠S ≅ ∠Q TR ≅ PR, RP ≅ RQ RT ≅ RS PQ ≅ TS
Prove: Δ P R Q ≅ ΔT R S
ΔPRQ ≅ ΔTRS under both (side-side-side) SSS condition and (angle-side-angle) ASA condition.
What exactly does "triangle congruency" mean?If all three corresponding sides are equal and all three corresponding angles are equal in measure, two triangles are said to be congruent.These triangles can be moved, rotated, flipped, and turned to look exactly the same.They will coincide if they are repositioned.Two triangles are congruent if they satisfy the five congruence conditions.They are the side-side-side (SSS), the side-angle-side (SAS), the angle-side-angle (ASA), the angle-angle-side (AAS), and the right angle-hypotenuse-side (RHS).
So,
Given: ∠P ≅ ∠T, ∠S ≅ ∠Q, TR ≅ PR, RP ≅ RQ, RT ≅ RS, PQ ≅ TS
To Prove: Δ PRQ ≅ ΔTRS
∠P ≅ ∠T∠S ≅ ∠QTR ≅ PRRP ≅ RQRT ≅ RSPQ ≅ TSSo, ΔPRQ ≅ ΔTRS under both SSS condition and ASA condition.
Therefore, ΔPRQ ≅ ΔTRS under both SSS condition and ASA condition.
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Please help me please help me
Answer: 1st option
Step-by-step explanation:
Black to Red is the transformation
C to C'
I to I'
Q to Q'
G to G'
1st, graph is moved up by 1 unit(+1 in x direction)
Then, graph is moved to right by 5 units(+5 in y direction)
(1, 5) is the change ==> 1st option
12 x (4 + 16) -78 x 3=
Please help me UwU
Answer:
The answer is 6.
Step-by-step explanation:
Simply by calculation.
need help solving please
Answer:
d
Step-by-step explanation:
Before it was demolished, the RCA Dome was home to the Indianapolis Colts. The attendance in 2001 was 450,746, and the attendance in 2005 was 457,373 .
b. If this rate of change continues, predict the attendance for 2012 .
If this rate of change continues, the attendance for the year 2012 will be 468973.
Slope(m) for given two points (x₁,y₁) and (x₂,y₂) is calculated using the formula
m=(y₂-y₁)/(x₂-x₁)
in the given question ,we will take year as x coordinate and the attendance as y coordinate.
Given " The attendance in 2001 was 450,746" , So
The point becomes (2001,450746)
and "the attendance in 2005 was 457,373" So
the point becomes (2005,457373)
Now as we have the two points , slope can be calculated as
m=(457373-450746)/(2005-2001)
m=6627/4
m≅1657
Hence the rate of change is 1657 persons per year
It is given in the question that rate of change remains constant.
To predict the attendance for the year 2012.
Let the attendance for the year 2012 be x.
we have (2001,450746) and (2012,x)
let m=1657, x₁=2001 , y₁=450746 , x₂=2012 , y₂=x
m=(y₂-y₁)/(x₂-x₁)
Substituting the values we get
[tex]1657=\frac{x-450746}{2012-2001} \\ \\ 1657=\frac{x-450746}{11}[/tex]
By cross multiplying we get
[tex]1657*11=x-450746\\ \\ 18227=x-450746\\ \\ x=450746+18227\\ \\ x=468973[/tex]
Therefore , The attendance for the year 2012 will be 468973.
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Rogelio has completed a total of 168 pull-ups in gym class this year. The number of pull-ups Rogelio has completed is 30% of the number of the pull-ups Armando has completed. What is the minimum number of additional pull-ups Rogelio must complete in order for his number of pull-ups to be greater than Armando's if Armando does not complete another pull-up all year?
Answer:
561
Step-by-step explanation:
168/3=56 (10 percent of armandos pull ups)
56 x 10 = 560 (100 percent of armandos pull ups)
Now it is equal. We are looking for greater than armando. You would need 561 pull ups to have more
The midpoint of QR is M(2, −5). One
endpoint is R(8, -1). Find the
coordinates of the other endpoint Q.
Write the coordinates as decimals or
integers.
Q=
Answer:
Q= ( -4 -9)
Step-by-step explanation:
Using the midpoint formula
X1 + X2 Y1 + Y1
----------, ---------
2. 2
The midpoint QR = (2, -5) and R = (-8, 1) Q =>(x, y)
= x + 8. y - 1
------------ = 2 and -------- = -5
2 2
=> (4 = x + 8) and( -10= y-1)
=> 4 -8 = x and -10+1= y
Q(x, y) = (-4, -9)
Function f. graphed below, is NOT an invertible function.
To which intervals could we restrict the domain off to make it an invertible function?
Choose all answers that apply:
the function f(x) is invertible on the intervals:
B: -7 ≤ x ≤ -4
C : 1 ≤ x ≤ 3
In which intervals is the function invertible?A function f(x) can only be invertible if it is one-to-one in its domain, where one-to-one means that each output is related to only one input.
So, if for two different values x₁ and x₂ we have that:
f(x₁) = f(x₂) = y, then the function is not invertible.
So here what we need to do is identify in which interval is the function one-to-one.
If you look at the last option, we have the interval [1, 3]
We have that f(1) = 2, and then the function increases until f(3) = 4
In that interval the function is cleraly one-to-one, so, on that interval, we can find an inverse function to f(x).
Similarly, on the second option [-7, -4] we also can see that:
f(-7) = 2
And then the function increases until f(-4) = 4
Then the function is also invertible on that interval.
We conclude that the function f(x) is invertible on the intervals:
B: -7 ≤ x ≤ -4
C : 1 ≤ x ≤ 3
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Answer:-3<x<-1 is incorrect, all else are correct.
Step-by-step explanation:
HELP
A person makes an overseas phone call. The telephone company charges 58 cents for the first minute and then 66 cents for each minute there after. Because it's an overseas phone call, there is also a 50c ent service charge. If the phone call cost $14.28, how many minutes did this person talk? The person talked for ____ minutes.
Answer:
Step-by-step explanation:
58 + 66(x-1) + 50 = 1428cents
66(x-1) + 108 = 1428
66(x-1) =1320
x-1 = 20
x = 21 mins
which number line represents the solution to the inequality -9x+9>36?
Answer:
x<-3
see the number line attached
make sure the number line has an open circle at -3
Hello,
[tex] - 9x + 9 > 36[/tex]
[tex] - 9x > 36 - 9[/tex]
[tex] - 9x > 27[/tex]
[tex]x < \frac{27}{ - 9 \:} \: \: because \: - 9 < 0[/tex]
[tex]x < - 3[/tex]
In 2006, a group of art museums collaborated to form the West Texas Triangle to promote their collections. This region is formed by the cities of Odessa, Albany, and San Angelo. The approximate coordinates of each location, respectively, are 31.9°N102.3°W,32.7°N99.3°W and 31.4°N100.5°W . Write a coordinate proof to prove that the West Texas Triangle is approximately isosceles.
We have proved that the West Texas Triangle formed is an isosceles triangle.
The approximate coordinates of three locations is given.
We have to prove that the West Texas Triangle is an isosceles triangle.
What is a triangle ?
A triangle is a 3-sided shape. There are three sides and three angles in every triangle, some of which may be the same.
As per the question ;
The West Texas Triangle is an isosceles triangle.
1st location has coordinates (31.9°N , 102.3°W)
2nd location has coordinates (32.7°N , 99.3°W)
3rd location has coordinates (31.4°N , 100.5°W)
Let's represent 1st location as A , 2nd location as B and 3rd as C.
Let's check it.
Using the distance formula ;
AB = √ [ (32.7 - 31.9) ² + (102.3 - 99.3)²]
AB = √ (0.8² + 3²)
AB = √9.64
AB = 3.10
Similarly ;
BC = √ (1.3² + 1.2²)
BC = √3.13
BC = 1.8
And
AC = √(0.5² + 1.8²)
AC = 1.8
The two sides are of equal length , so , we can conclude that this is an isosceles triangle.
Thus , the West Texas Triangle formed is an isosceles triangle.
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A local grocery store sells 32.5 oz bags of mixed nuts that contain 56% peanuts. To make their product they combine Planters mixed nuts which contain 50% peanuts and Emerald mixed nuts which contain 60% peanuts. How much of each do they need to use
The local grocery store mixed 13 oz of planters mixed nuts with 19.5 oz of emerald mixed nuts to get 32.5 oz bags of mixed nuts that contain 56% peanuts.
What is an equation?An equation is an expression that shows the relationship between two or more numbers and variables.
Let x represent the amount of planters mixed nuts and y represent the amount of emerald mixed nuts, hence:
x + y = 32.5 (1)
Also:
0.5x + 0.6y = 0.56(32.5) (2)
From both equations:
x = 13, y = 19.5
The local grocery store mixed 13 oz of planters mixed nuts with 19.5 oz of emerald mixed nuts to get 32.5 oz bags of mixed nuts that contain 56% peanuts.
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solve solve log3 (3x-2) - log3 (x+2) = 2
The solution to the logarithmic equation is - 10 / 3.
How to simplify a logarithmic equation to solve for a variable
Logarithms are examples of trascendent functions, that is, function that cannot be described in algebraic terms. Herein we must take advantage of logarithm properties to solve for x in the expression given. The complete procedure is shown below:
㏒₃ (3 · x - 2) - ㏒₃ (x + 2) = 2 Given
㏒₃ [(3 · x - 2) / (x + 2)] = 2 Logarithm of a division of functions
(3 · x - 2) / (x + 2) = 3² Definitions of power and logarithm
(3 · x - 2) / (x + 2) = 9 Definition of power
3 · x - 2 = 9 · (x + 2) Definition of division / Compatibility with multiplication / Existence of the multiplicative inverse / Associative and modulative properties
3 · x - 2 = 9 · x + 18 Distributive property
- 20 = 6 · x Compatibility with addition / Existence of the additive inverse / Associative, commutative and modulative properties
6 · x = - 20 Symmetric property
x = - 20 / 6 Definition of division / Compatibility of multiplication / Existence of the multiplicative inverse / Associative, commutative and modulative properties
x = - 10 / 3 Simplification of fractions / Result
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Determine a base and a height for each shap
True or False?
4 and 1/4 are opposites
Answer:
I want to say False.
Step-by-step explanation:
I would think that the opposite of 4 would be -4.
Good luck! :)
Answer:
True
Step-by-step explanation:
4 = 4/1 1/4 is the reciprocal or the opposite of 4
Find the area of shaded region in the given figure. if base 6cm hight 12 cm and side 7 cm
Answer: Area, S = 36 [tex]cm^{2}[/tex]
Step-by-step explanation:
What is the Area of a Triangle?
The area of a triangle is defined as the total region that is enclosed by the three sides of any particular triangle. Basically, it is equal to half of the base times height, i.e.
A = 1/2 × b × h.
Given data,
Area of shaded triangle S = 1/2 *b*h
We can write,
Area = S
Base = b
Height = h
we get,
Base b = 6cm
Height h = 12cm
h=12 cm height of isosceles triangle ,
Area = 1/2 * b*h
We can substitute b and h values,
Area= 1/2* 6 * 12
= 1/2 * 72
= 36
So, Area, S = 36 [tex]cm^{2}[/tex]
Therefore,
Area, S = 36 [tex]cm^{2}[/tex]
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Find the slope and y -intercept of each line.
-Ax+B y=-C
The slope of the given equation of line is [tex]\frac{A}{B}[/tex] and the y-intercept is [tex]\frac{-C}{B}[/tex]
What is the equation of a line to the curve?A line's equation has the standard form ax + by + c = 0. Here, the variables are x and y, the coefficients are a and b, and the constant term is c. It is a first-order equation with the variables x and y. The coordinates of the point on the line shown in the coordinate plane are represented by the values of x and y.Given: -Ax + By = -C
=> y = [tex]\frac{A}{B}x+ \frac{-C}{B}[/tex]
As we know that the slope-intercept form of an equation of a line is given by: y = mx + c, where:
m = slope of the linec = y-intercept of the line.On comparing the given equation of the line with this form, we get:
m = [tex]\frac{A}{B}[/tex] and c = [tex]\frac{-C}{B}[/tex]
Hence, The slope of the given equation of line is [tex]\frac{A}{B}[/tex] and the y-intercept is [tex]\frac{-C}{B}[/tex].
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Tetsuya and Mason want to determine the probability that a red marble will be chosen out of a bag of 4 red, 7 blue, 5 green, and 2 purple marbles. Is either of them correct? Explain your reasoning.
Both Tetsuya and Mason are interested in finding the probability that a red marble will be chosen out of the bag. We will decide if either of them is correct.
In statistics, what does a probability mean?
The probability serves as a gauge for how likely an event is to occur. It gauges how likely an event is. P(E) = Number of Favorable Outcomes/Number of Total Outcomes is the formula for probability.Tetsuya P( R) = 4/17
Mason P( R ) = 1 - 4/18
P ( R) = Favorable outcome/ possible outcome
The number of favorable outcomes is the number of red marbles in the bag, so it is equal to 4. The number of possible outcomes is the number of all marbles in the bag. Let's find it.
4 + 7 + 5 + 2 = 18
We have enough information to find the probability that a red marble will be chosen P(R).
P( R) = 4/18
Tetsuya used the wrong number of possible outcomes in the denominator. Let's look at Mason's solution. They wanted to use the probability of complementary events. However, they found the probability that a selected marble is not red.
Mason P( R ) = 1 - 4/18
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Carolyn needs to figure out the missing properties used to simplify the expression below
Expression Property used
Step 1 1 x 3/2 x (-5) x 3 x (-4) Inverse Property of Multiplication
Step 2 1 x (-4) x 3/2 x (-5) x 3 Associative Property of Multiplication
Step 3 1 x [ (-4) x 3/2 ] x (-5) x 3 Commutative Property of Multiplication
Step 4 1 x [-6] x (-5) x 3 Identity Property of Multiplication
Step 5 -6 x (-5) x 3
What is Inverse property of multiplication?The inverse property of multiplication refers to multiplying a number with it's reciprocal. This is applied while transforming from division to multiplication as division is the inverse of multiplication.
What is Associative Property of Multiplication?Associative property of multiplication is applied while multiplying several values, Hence rearranging the terms will still give same answer.
What is Commutative Property of Multiplication?This property has it that the order of the multiplication do not matter, hence multiplication can start with any order.
What is Identity Property of Multiplication?This property refers to multiplication of a value with one, such multiplication usually return same value as the product.
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please factor 27x^6 y^9 - 64 and please show me the steps to complete it
Answer:
Hello,
Step-by-step explanation:
Do you remember the formula
[tex]\boxed{a^3-b^3=(a-b)(a^2+ab+b^2)}\\\\\\27x^6y^9-64=(3x^2y^3)^3-4^3\\\\=(3x^2y^3-4)((3x^2y^3)^2+(3x^2y^3*4)+4^2)\\\\\\=(3x^2y^3-4)(9x^4y^6+12x^2y^3+16)\\[/tex]
Find the geometric mean between pair of numbers.
8√2/3 and 4√2/3
The geometric mean between pair of numbers 8√2/3 and 4√2/3 is 2.67
We know that for a sequence of numbers x1, x2,.., xn
The formula of geometric mean is,
[tex]\Pi=\sqrt[n]{x_1.x_2.x_3...x_n}[/tex]
In this question, we have been given two numbers 8√2/3 and 4√2/3
We need to find the geometric mean between pair of numbers, 8√2/3 and 4√2/3.
Using the formula for the geometric mean,
[tex]\Pi=\sqrt[n]{x_1.x_2.x_3...x_n}[/tex]
In this question, we have two numbers.
So, the value of n is 2.
[tex]\Pi=\sqrt{\frac{8\sqrt{2} }{3} \times \frac{4\sqrt{2} }{3}} \\\\\Pi=\sqrt{\frac{32\times 2}{9} } \\\\\Pi=\sqrt{\frac{64}{9} } \\\\\Pi=\frac{8}{3}\\\\\Pi=2.67[/tex]
Therefore, the geometric mean between pair of numbers 8√2/3 and 4√2/3 is 2.67
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what is 3x5x3x4x5x2 =
(GIVING BRAINLIST)
Answer:
1800
Step-by-step explanation:
hope this works for you :)
