An infinite geometric series is the outcome of an unlimited geometric sequence. If the value of S = 12 and r = 1/6 then the value of a₁ = 10.
What is an infinite geometric series?An infinite geometric series is the outcome of an unlimited geometric sequence. This series wouldn't have a finish. It is possible to calculate the sum of all finite geometric series. The terms in the sequence will grow steadily larger, but adding the larger numbers together will not provide a solution if the common ratio of an infinite geometric series is greater than one.
The sum of an infinite geometric series be S = a/1-r
S be the sum = 12
a₁ be the first term = ?
r -be the common ratio = 1/6
We know,
S = a₁/1-r
substitute the values in the above equation, we get
12 = a₁/1-(1/6)
simplifying the equation,
12 = 6a₁/5
a₁ = 60/6 = 10
If the value of S = 12 and r = 1/6 then the value of a₁ = 10.
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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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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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describe the transformation then graph f(x)=x+4; g(x)=f(x-2)
Answer:
Use This it will help
Step-by-step explanation:
What is the answer to this problem ?
Answer:
2(x + 6) = 38
Step-by-step explanation:
First, we choose a variable to represent the unknown number.
1. Let x = the unknown number.
Then, we translate the sentence into an equation using the variable we chose.
"Twice the sum of six and a number is 38."
We do it piece by piece:
"the sum of six and a number": x + 6
"Twice the sum of six and a number": 2(x + 6)
"Twice the sum of six and a number is 38.": 2(x + 6) = 38
Now we have an equation that represents the sentence.
2(x + 6) = 38
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)
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.
The scale of the drawing was 1 inch : 5 yards If the actual width of a naborhood park is 70 yards how wide is the park in the drawing
Based on the scale of the drawing, and the actual width, the width of a park in a drawing would be 14 inches.
What is the width?A single inch of the width of a park in a drawing, is 5 yards of the width of the actual park.
This means that a single inch equates to 5 yards.
If there were therefore 70 yards in the actual park, the width of the drawing is:
= 70 / 5
= 14 inches
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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
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]
prove (1+cot55) (1+cot80) (1+cot77) (1+cot58)=4
cot(55°) = 0.70021 ... cot(58°) = 0.62487 ... cot(77°) = 0.23087 ... cot(80°) = 0.17633
(1+cot55)(1+cot80)(1+cot77)(1+cot58).
In trigonometry, the cot or cotangent is one of the six trigonometric ratios. In a right triangle, the lattice of angles is equal to the ratio of the adjacent and opposite sides of the angle. Cot x is also equal to the reciprocal of tan x.
Therefore, the cotangent of the angle α of a right triangle is equal to the length of the adjacent side b divided by the opposite side a. To remove the crib, simply enter the length of the adjacent and opposite sides, then remove it. This formula can be very similar to the formula used to calculate the tangent.
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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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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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Determine a base and a height for each shap
Which set of numbers cannot be the measures of the sides of a triangle?
A 10,11,20
B 14,16,28
C 35,45,75
D 41,55,98
Option D is the correct answer.
Triangle Inequality Theorem states that the sum of two side lengths of a triangle is always greater than the third side.
If this is true for all three combinations of three combinations of side lengths.
i.e., a + b > c , b + c > a and c + a > b then the length form a triangle.
(A) 10 + 11 = 21 > 20
And 11 + 20 =31 > 10
And, 20 + 10 = 30 > 11
So, it forms a triangle
(B) 14 + 16 = 30 >28
And 16 + 28 = 44 >14
And 28 + 14 = 42 > 16
So, it forms a triangle
(C) 35 + 45 = 80 > 75
And 45 + 75 = 120 > 35
And 75 + 35 = 110 > 45
So, it forms a triangle
(D) 41 + 55 = 96 < 98
It does not form a triangle
because sum of two sides lengths is less than the third side length.
Hence, Option D is the correct answer
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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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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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What is the width of the prism?
4 ft
w
2 ft
Volume = 24 cubic feet
Answer: 3 ft
Step-by-step explanation:
l=length. w=width. h=height. V=volume
V=l*w*h
24=4*w*2
24=8*w
8w=24
w=3 ft
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
F four out of every nine trick-or-treaters that came to your house last halloween were dressed as snow white, what proportion of trick-or-treaters were not dressed as snow white?
The ratio of trick-or-treaters who were not dressed as snow white last Halloween is 5/9
Ratio is the relationship between two quantities, normally expressed as the quotient of one divided by the other.
It can be expressed in different ways:
1) Use the ":" to separate the values : a : b
2) Use the word "to" : a to b
3) Write it in fraction: a / b
On the other hand, proportion is the equality of two or more ratios.
example: a : b = c : d
If four out of every nine trick-or-treaters that came to your house last Halloween were dressed as snow white, then the rest were not dressed as snow white.
9 - 4 = 5
If five out every nine trick-or-treaters that came to your house last Halloween were not dressed as snow white, we can express the ratio as
5 out of 9 or 5 : 9 or 5/9
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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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Does the infinite series converge or diverge? If it converges, what is the sum?
b. 1/3 - 1/9 + 1/27 - 1/81 + . . . .
A geometric series exist as the totality of an infinite number of terms that contain a constant ratio between consecutive terms.
The sum of the infinite series is 1/4.
What is an infinite geometric series?The result of an endless geometric sequence exists an infinite geometric series. There would be no conclusion to this series. There would be no conclusion to this series. The infinite geometric series has the general form a1 + a1r + a1r² + a1r³+…, where r is the common ratio and a1 is the first term.
The equation be S = a1/(1-r), where s is the total, a1 is the series first term, and r is the common ratio, is a general formula for calculating the sum of an infinite geometric series. The formula a2/a1, where a2 is the second term in the series and a1 is the first term in the series, can be used to determine the common ratio.
Given: a(first term) = 1/3
r(common ratio) = a2/a1
= (-1/9)/(1/3) = -1/3
Since | r | ∠ 1, the infinite series converges.
The sum of infinite geometric series is:
S∞ = a/(1 - r); -1 ∠ r ∠1
S∞ = (1/3) / (1+(1/3)) = 1/4
The sum of the infinite series is 1/4.
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The points (-2, 8) and (r, − 1) lie on a line with slope -3. Find the missing coordinate r.
ILL GIVE BRAINLIEST TO WHOEVER HELPS ME QUICKLY AND CORRECTLY!
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 two solutions to the equation below.
(x-7)(x - 5) = 0
Answer: x = 5, 7
Step-by-step explanation:
This equation is already factored for us, so we can split it into two separate equations set equal to 0 and solve for x.
Given:
(x - 7)(x - 5) = 0
Equation #1:
x - 7 = 0
x = 7
Equation #2:
x - 5 = 0
x = 5
Solution:
x = 5, 7
Answer:
x = 5, x = 7
Step-by-step explanation:
(x - 7)(x - 5) = 0 ← in standard factored form
equate each factor to zero and solve for x
x - 7 = 0 ⇒ x = 7
x - 5 = 0 ⇒ x = 5
Express the interval using two different representations.
x<-11 \text{ or } x>0
x<−11 or x>0
The interval using two different representations are (-∝, 11) ∪ (0, ∝) and {x | x < −11 or x > 0}
How to express the interval using two different representations?The interval is given as
x < −11 or x > 0
Inequalities like < and > are represented using the parenthesis ( and )
Also, the term "or" mean union and it is represented using ∪
So, one representation of the interval can be represented as
(-∝, 11) ∪ (0, ∝)
Another representation of the interval is {x | x < −11 or x > 0}
Hence, the interval using two different representations are (-∝, 11) ∪ (0, ∝) and {x | x < −11 or x > 0}
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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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Solve: x+2/x-4<0
A –4 < x < –2
B –2 < x < 4
C –2 < x < –4
D –4 < x < 2
Answer:
B. -2 < x < 4
Step-by-step explanation:
The function is (x +2)/(x-4) and we are asked to determine the x interval when the inequality (x +2)/(x-4) < 0 is satisfied
1. Find the critical points
The critical points are
x = - 2 when the numerator becomes -2 + 2 = 0, and f(x) = 0
and
x = 4 when the denominator becomes 4 - 4 = 0 and (x + 2)/0 is undefined
So critical points are -2 and 4
2. Use these critical points to divide the number line into intervals
The intervals are -∞ < x < -2, -2 < x < 4 and 4 < x < ∞
these can be re-written as x < -2, -2 < x < -4 and x > 4
3. For each of these intervals take a test point and see if it satisfies that inequality
(a) -∞ < x < -2 Take x = -3 which is less than -2
We get f(-3) = (-3 + 2)/(-3 -4) = -1/-7 which is + 1/7 and does not satisfy the (x+2)/(x-4) < 0 inequality.
So we can discard this interval
(b) For interval -2 < x < 4, Let's take x = 0
(0+2)/(0-4) = 2/(-4) = -1/2 = -0.5 which does satisfy the inequality
(x+2)/(x-4) <0, so this is a valid interval
(c) For interval x > 4, take x = 5
(x+2)/(x-4 ) = (5 + 2)/(5-4) = 7/1 = 7 and does not satisfy (x+2)/(x-4 ) < 0
So the interval which satisfies the inequality is -2 < x < 4
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
Evaluate each expression for x=-2,0,1 , and 4.
3x+1
The value for x = -2,0,1 and 4 is -5,1,4,13 respectively.
What does finding x's value mean?
To "solve for x" is to determine the value of x that would result in the equation being true. Think about the following equation: x plus 1 = 3. You would have to figure out a value for x such that it produces three when one is added in order to try to solve it.
For x = -2,
3x+1
= 3(-2)+1
= -6 +1
= -5
For x = 0
3x+1
= 3(0) + 1
= 1
For x = 1
3x+1
= 3(1) + 1
= 4
For x = 4
3x+1
= 3(4) + 1
= 13
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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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