for Given:
[tex]\begin{gathered} a_1=5 \\ r=-2 \end{gathered}[/tex]You need to remember that "r" is the Common ratio between the terms of the Geometric Sequence and this is the first term:
[tex]a_1_{}_{}[/tex]The formula the nth term of a Geometric Sequence is:
[tex]a_n=a_1\cdot r^{(n-1)}[/tex]Where "n" is the number of the term, "r" is the Common Ratio, and the first term of the sequence is:
[tex]a_1[/tex]In this case, since you need to find the 8th term, you know that:
[tex]n=8[/tex]Then, you can substitute all the values into the formula:
[tex]a_8=(5)(-2)^{(8-1)}[/tex]Evaluating, you get:
[tex]\begin{gathered} a_8=(5)(-2)^{(7)} \\ a_8=(5)(-128) \\ a_8=-640 \end{gathered}[/tex]Hence, the answer is:
[tex]a_8=-640[/tex]
use the order of operations to find the value of the following expression
If a regular polygon has exteriorangles that measure approximately17.14° each, how many sides doesthe polygon have?
To answer this question we will set and solve an equation.
Recall that the exterior angle of an n-gon has a measure of:
[tex]\frac{360^{\circ}}{n}.[/tex]Let n be the number of sides that the polygon that we are looking for has. Since the regular polygon exterior angles with a measure of approximately 17.14 degrees, then:
[tex]\frac{360^{\circ}}{n}\approx17.14^{\circ}.[/tex]Therefore:
[tex]n\approx\frac{360^{\circ}}{17.14^{\circ}}[/tex]Simplifying the above result we get:
[tex]n\approx21.[/tex]Answer: 21 sides.
The famous mathematician Gauss is credited with deriving a formula for determining the the sum of the first n counting numbers. If the sum of the first 100 counting numbers is 5050, what is the difference between the sum of all of the even counting numbers and the odd counting numbers less than 101? Start by making the problem simpler and look for patterns. Describe how you came to your solution.
Given:
The sum of the first 100 counting numbers is 5050.
To find:
The difference between the sum of all of the even counting numbers and the odd counting numbers less than 101.
Explanation:
Let us find the sum of all of the even counting numbers from 1 to 101.
The series is,
[tex]S_1=2+4+6+....+100[/tex]It can be written as,
[tex]S_1=2(1+2+3+.....+50)[/tex]Using the formula,
[tex]\begin{gathered} 1+2+3+.....+n=\frac{n(n+1)}{2} \\ S_1=2(1+2+3+....+50)=2[\frac{50(50+1)}{2}] \\ S_1=50(51) \\ S_1=2550........(1) \end{gathered}[/tex]Next, let us find the sum of all of the odd counting numbers.
[tex]\begin{gathered} S_2=Total-Sum\text{ of all even numebrs} \\ S_2=5050-2550 \\ S_2=2500.......(2) \end{gathered}[/tex]So, the difference between the sum of all of the even counting numbers and the odd counting numbers less than 101 is
[tex]\begin{gathered} S_1-S_2=2550-2500 \\ =50 \end{gathered}[/tex]Final answer:
The difference between the sum of all of the even counting numbers and the odd counting numbers less than 101 is 50.
what's the probability of randomly meeting a four child family with either exactly one or exactly two boy children
1) Let the Probability of randomly meeting a four child family with exactly one child: P(A)
Let the Probability of randomly meeting a four child family with exactly 2 boy children : P(B)
Since the question is about how do we get to the Probability of meeting A or B
We can write:
P(A ∪ B) = P(A) + P(B) - P(A * B)
2) Knowing the subspace. We subtract to not count twice the Probability of A , and B.
If the events are mutually exclusives, i.e. there are no common elements so so we can write that
P(A ∪ B)= P(A) +P(B)
What is the equation of the line that passes through the point (-5, -3) and
has a slope of -3/5?
Answer:
y = (-3/5)x - 6
Step-by-step explanation:
m = slope: (-3/5); (-5, -3)
(x₁, y₁)
y - y₁ = m(x - x₁)
y - (-3) = (-3/5)(x - (-5)
y + 3 = (-3/5)(x + 5)
y + 3 = (-3/5)x - 3
-3 -3
-------------------------
y = (-3/5)x - 6
I hope this helps!
In the diagram below, AB is a diameter of the circle. If arc CB measures 98 °, find the measure of < ABC.
In this problem
arc ACB=180 degrees -----> because AB is a diameter
arc ACB=arc AC+ arc CB ----> by addition angles postulate
substitute given values
180=arc AC+98
arc AC=82 degrees
Find out the measure of angle mm by inscribed angle
mm
The answer is option AThis statement is false or true?Expression that contain one variable can be proven true or false by replacing the variable with a number.
The statement is false.
An expression has no value of true since it is not an equation.
Find the slope of the line that passes through (-31, 26) and (4, 36).
The slope of a line can be calculated with the following formula:
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]You know that the line passes through the following points:
[tex]\mleft(-31,26\mright);(4,36)[/tex]For this case, you can set up that:
[tex]\begin{gathered} y_2=36 \\ y_1=26 \\ x_2=4 \\ x_1=-31 \end{gathered}[/tex]Then, knowing the coordinates shown above, you can substitute them into the formula in order to find the slope of the line. This is:
[tex]\begin{gathered} m=\frac{36-26}{4-(-31)} \\ \\ m=\frac{10}{35} \\ \\ m=\frac{2}{7} \end{gathered}[/tex]The answer is:
[tex]m=\frac{2}{7}[/tex]Can you help me with this and break it down if you can ?
Given:
[tex]\begin{gathered} y=3x^2\text{ + 13x -50} \\ y\text{ = 13x }-\text{ 2} \end{gathered}[/tex]Subtracting equation 2 from 1:
[tex]\begin{gathered} y-y\text{ = }3x^2\text{ + 13x - 50 -(13x - 2)} \\ 0=3x^2\text{ + 13x - 50 - 13x + 2} \\ 3x^2\text{ -48 = 0} \end{gathered}[/tex]Solving for x:
[tex]\begin{gathered} 3x^2\text{ - 48 = 0} \\ 3x^2\text{ = 48} \\ \text{Divide both sides by 3} \\ x^2\text{ = }\frac{48}{3} \\ x^2\text{ = 16} \\ \text{Square root both sides} \\ x\text{ = }\sqrt[]{16} \\ x\text{ = }\pm\text{ 4} \end{gathered}[/tex]Substituting the value of x into equation 2:
[tex]\begin{gathered} y\text{ = 13x - 2} \\ y\text{ = 13(}\pm4)\text{ - 2} \\ y\text{ = 52 - 2 } \\ =\text{ 50} \\ or\text{ } \\ y\text{ = -52 - 2} \\ =\text{ -54} \end{gathered}[/tex]Hence, the solution to the system of equations is:
(4, 50) and (-4 , -54)
Translate the sentence into an inequality.The sum of a number times 6 and 18 is at least -28.Use the variable b for the unknown number.
Traslating the sentence into an inequality, we get:
[tex]6b+18\ge-28[/tex][tex] \sqrt{16} [/tex]can you do a step by step explanation to find the square root.
Explanation
Step 1
a square root is given by:
[tex]\begin{gathered} \sqrt[]{a}=b \\ \text{where} \\ b^2=a \end{gathered}[/tex]look for values for b
[tex]undefined[/tex]-sqrt-50 in radical form
We have the following expression:
[tex]-\sqrt[]{-50}[/tex]The prime factorization of 50 is
[tex]\begin{gathered} 50=2\times5\times5 \\ 50=2\times5^2 \end{gathered}[/tex]Then, we can rewritte our expression as
[tex]-\sqrt[]{-50}=-\sqrt[]{-(2\times5^2})=-i\sqrt[]{2\times5^2}[/tex]because the square root of -1 is defined as the complex i. Then, we have
[tex]\begin{gathered} -\sqrt[]{-50}=-i\times\sqrt[]{2}\times\sqrt[]{5^2} \\ or\text{ equivalently,} \\ -\sqrt[]{-50}=-i\times\sqrt[]{2}\times5 \end{gathered}[/tex]Therefore, the answer is
[tex]-\sqrt[]{-50}=-5\sqrt[]{2}\text{ i}[/tex]nes ing Online book David's dad drove at a constant rate for 25 miles. It took him 20 minutes. At what rate was David's dad driving (in miles per hour)? 55 miles per hour 65 miles per hour 75 miles per hour ps 85 miles per hour #
In order to calculate the rate (that is, the speed) David's dad was driving in miles per hour, first let's convert the time from minutes to hours using a rule of three:
[tex]\begin{gathered} 1\text{ hour}\to60\text{ minutes} \\ x\text{ hours}\to20\text{ minutes} \\ \\ 60x=20\cdot1 \\ x=\frac{20}{60}=\frac{1}{3} \end{gathered}[/tex]Now, to find the speed, we just need to divide the distance by the time:
[tex]\text{speed}=\frac{25}{\frac{1}{3}}=25\cdot3=75\text{ mph}[/tex]So the speed is 75 mph, therefore the answer is the third option.
2/9 + 4/9 ..........
We will do the operation:
[tex]\frac{2}{9}+\frac{4}{9}[/tex]As both fractions have the same denominator, we add the numerators, and we obtain:
[tex]\frac{2}{9}+\frac{4}{9}=\frac{6}{9}=\frac{2}{3}[/tex]Where we simplified 6/9 to 2/3 by dividing by 2.
This means that 2/9+4/9 is 2/3.
Help please look at the image and also use these terms recursive: f(1) = 2, f(n) = 2*f(n-1). explicit: we need to take 1st term/pattern.
The explicit formula for a geometric sequence is given by:
[tex]f(n)=f(1)r^{n-1}[/tex]where r is the common ratio of the sequence.
For this sequence the common ratio is 2 and the first term is 2, therefore its explicit formula is:
[tex]f(n)=2(2)^{n-1}[/tex]The recursive formula for a geometric sequence is given by:
[tex]\begin{gathered} f(1) \\ f(n)=rf(n-1) \end{gathered}[/tex]Therefore in this case we have:
[tex]\begin{gathered} f(1)=2 \\ f(n)=2f(n-1) \end{gathered}[/tex]please help me ASAP!!!
1)
The expression :
[tex]\begin{gathered} 2^3\cdot2^5=2^8 \\ \text{Tha base are same so, the exponents are add up} \end{gathered}[/tex]1-same Base Product
2)
The expression:
[tex]\begin{gathered} \frac{5^5}{5^2}=5^3 \\ \text{The base are same and they are divison from so the exponents will subtract} \\ \end{gathered}[/tex]2- Same base Quotient
3)
The expression:
[tex]\begin{gathered} (3^2)^3=3^6 \\ \text{The }power\text{ to pwer will multiply sor 2}\times3=6 \end{gathered}[/tex]3-Power to power
4)
The expression:
[tex]\begin{gathered} 8^0=1 \\ The\text{ zero power is always equal to 1} \end{gathered}[/tex]4- Zero power
Answer:
1) same Base Product
2) Same base Quotient
3) Power to power
4) Zero Power
I still can’t get a hold of questions like this.
We are given that a job pays 8% of the sales. Let's say that "x" is the amount sold per week. Then the payment for a week ales
I’m having trouble with this calculus practice problem Below are the answer options A. -2B. 1C. 3D. The limit does NOT exist
Given
[tex]\lim _{x\to-3^+}h(x)[/tex]Solution
The limit is tending to -3 from the right, that is why it is written as
[tex]-3^+[/tex]From the graph, we will trace the graph from the right to -3
The final answer is -2
a line that passes through points (2, 40) and (20, 4)
Answer
y - 40 = -2 (x - 2)
We can simplify this
y - 40 = -2x + 4
y = -2x + 4 + 40
y = -2x + 44
Explanation
The general form of the equation in point-slope form is
y - y₁ = m (x - x₁)
where
y = y-coordinate of a point on the line.
y₁ = This refers to the y-coordinate of a given point on the line
m = slope of the line.
x = x-coordinate of the point on the line whose y-coordinate is y.
x₁ = x-coordinate of the given point on the line
We need to calculate the slope and to use one of the points given as (x₁, y₁)
For a straight line, the slope of the line can be obtained when the coordinates of two points on the line are known. If the coordinates are (x₁, y₁) and (x₂, y₂), the slope is given as
[tex]Slope=m=\frac{Change\text{ in y}}{Change\text{ in x}}=\frac{y_2-y_1}{x_2-x_1}[/tex](x₁, y₁) and (x₂, y₂) are (2, 40) and (20, 4)
[tex]\text{Slope = }\frac{4-40}{20-2}=\frac{-36}{18}=-2[/tex]Slope = m = -2
(x₁, y₁) = (2, 40)
x₁ = 2, y₁ = 40
y - y₁ = m (x - x₁)
y - 40 = -2 (x - 2)
We can simplify this
y - 40 = -2x + 4
y = -2x + 4 + 40
y = -2x + 44
Hope this Helps!!!
if you could draw the graph, that would be great!!
The functions we have are:
[tex]\begin{gathered} F(x)=x^2 \\ G(x)=3x+1 \end{gathered}[/tex]And we need to graph F-G
Step 1. Find the expression for F-G.
We subtract the expressions for F(x) and G(x):
[tex]F-G=x^2-(3x+1)[/tex]Simplifying this expression:
[tex]F-G=x^2-3x-1[/tex]Step 2. Graph the expression.
In the following image, we can thee the graph for F-G:
Hi, I have no clue how to do proportions and can you explain how to do this? If you can't that's alright.
___________________
Please, give me some minutes to take over your question
______________________________________
Rate = miles / time
8/t = 7/ 35
Dividing by 7
8/t = 7/ 35
8/ 7t = 1/ 35
Multiplying by t
8/7 = t/35
_____________
Options
1) 8/t = 35/ 7 (False, t/8 = 35/ 7 )
2) t/8 = 7/ 35 (False, t/8 = 35/ 7 )
3) 8/7 = t/ 35 (TRUE)
4) 7/8 = t/35 (False, 8/7 = t/ 35 )
__________________
Answer
3) 8/7 = t/ 35 (TRUE)
Find the Area of the figure below. Round to the nearest tenths place
The Figure contains a trapezium and a semicircle. The area of the figure would be the sum of the area of the trapezium and the area of the semicircle. The formula for finding the area of a trapezium is expressed as
Area = 1/2(a + b)h
where
a and b are the length of the parallel sides of the trapezium
h = height of trapezium
From the diagram,
a = 13
b = 6
h = 8
Area = 1/2(13 + 6)8
Area = 76
The formula for finding the area of a semicircle is expressed as
Area = 1/2 x pi x radius^2
pi = 3.14
diameter = 6
radius = diameter/2 = 6/2
radius = 3
Area = 1/2 x 3.14 x 3^2
Area = 14.13
Area of figure = 76 + 14.13
Area of figure = 90.1
the picture shows the graphing numbers here are the questions: b. how much does the investment grow every year?c. how much money did the investment start out as?d. what sequence equation would represent this graph?e. hat would the value of the investment be after another 10 years?f. what would the value of the investment be after a total of 20 years.
Part b) the trick consists of noting that the difference between the investment of any two consecutive years is the same: $1,750. (In general, this kind of table is called an arithmetic sequence). How much does the investment grow every year? Exactly $1,750.
Part c) The idea here is to find the "first term", which is the investment when everything began (first year): $20,000. (this could seem trivial, but it will be important).
Part d) Remember I told you that this kind of table is called arithmetic sequence (a_n). This means that they have the general (generic) form:
[tex]a_n=\text{ initial value}+(n-1)\cdot\text{ (growing rate)}[/tex]By part b and c, our initial value is $20,000 and our growing rate is $1,750. So we get
[tex]a_n=20000+(n-1)\cdot1750[/tex]Comment: You can think that those dates (initial term, and growing rate) are all you need to understand this kind of table.
Part e) This type of question reveals the "power" of the formula we obtained above (now we can make projections regarding the future; namely, beyond the table).
Now, there is a detail to keep in mind; the wording "another 10 years". It means we must find the value of the sequence in 15, not 10.
[tex]a_{15}=20000+(15-1)\cdot1750=44500[/tex]Part f) Here there is no trick; we just need to calculate the 20th term of the sequence:
[tex]a_{20}=20000+(20-1)\cdot1750=53250[/tex]
Drag each label to the correct location. Not all labels will be used.The dimensions of a rectangular section of forest land are 5.5 x 105 meters and 4.2 x 104 meters. Complete the following sentences.2.31 x 1032.31 x 1042.31 x 10523.1 x 102.31 < 101023.1 x 1010square meterssquare kilometersThe area of the land issquare meters in scientific notation.We can represent this area assquare kilometers in scientific notation.Hint: 1 square kilometer is equal to 1 x 106 square meters.The unitis more appropriate to represent the area of the forest land in scientific notation.
The area of the land would be (4.2x10^4)(5.5x10^5)=23.1x10^9
and we can represent this area in scientific notation like: 2.31x10^10
the unit more appropriated for the area is: square kilometers
Average movie prices in the unites States are, in general, lower than in other countries. it would cost $77.94 to buy three tickets in Japan plus two tickets in Switzerland. Three tickets in Switzerland plus two tickets in Japan would cost $73.86. How much does an average movie ticket cost in each countires?Japan average:Switzerland average:
If "J" is the average price in Japan and "S" is the average price is "S", then since we are told that three tickets in Japan plus two tickets in Switzerland cost $77.94 we have the following relationship:
[tex]3J+2S=77.94,\text{ (1)}[/tex]We are also told that three tickets in Switzerland plus two tickets in Japan would cost $73.86. This gives us the following equation:
[tex]2J+3S=73.86,(2)[/tex]We get two equations with two variables. To solve this system we will multiply equation (1) by -2:
[tex]-6J-4S=-155.88,(3)[/tex]Now we multiply equation (2) by 3:
[tex]6J+9S=221.58,(4)[/tex]Now we will add equation (3) and equation (4):
[tex]-6J-4S+6J+9S=-155.88+221.58[/tex]Now we add like terms;
[tex]5S=65.7[/tex]Dividing both sides by 5:
[tex]S=\frac{65.7}{5}=13.14[/tex]Now we replace the value of S in equation (1):
[tex]3J+2(13.14)=77.94[/tex]Solving the operation:
[tex]3J+26.28=77.94[/tex]Subtracting 26.28 to both sides:
[tex]\begin{gathered} 3J=77.94-26.28 \\ 3J=51.66 \end{gathered}[/tex]Dividing both sides by 3:
[tex]J=\frac{51.66}{3}=17.22[/tex]Therefore, the average in Japan is $17.22 and the average in Switzerland is $13.14.
2 (4k + 3)- 13 = 2 (18 - k) 13
Given the expression:
[tex]2(4k+3)-13=2(18-k)-13[/tex]solve for k :
[tex]2\cdot4k+2\cdot3-13=2\cdot18-2k-13[/tex][tex]8k+6-13=36-2k-13[/tex]combine the like terms:
[tex]undefined[/tex]What is the simplified form of each expression?a. 10^8b. (0.2)^5
Answer:
(a)100,000,000
(b)0.00032
Explanation:
(a)To determine the simplified form of 10^8
[tex]\begin{gathered} 10^8=10\times10\times10\times10\times10\times10\times10\times10 \\ =100,000,000 \end{gathered}[/tex](b)To determine the simplified form of (0.2)^5
[tex]\begin{gathered} 0.2^5=(0.1\times2)^5 \\ =0.1^5\times2^5 \\ =0.00001\times32 \\ =0.00032 \end{gathered}[/tex]Determine the equation of the line that passes through the point (1/9,−3) and is parallel to the line −8y+4x=4.
Given:
The point lies on the line is (1/9, -3).
The parallel line is -8y+4x=4.
Required:
We need to find the equation of the line.
Explanation:
Consider the parallel line.
[tex]-8y+4x=4[/tex]Subtract 4x from both sides.
[tex]-8y+4x-4x=4-4x[/tex][tex]-8y=4-4x[/tex]Divide both sides by (-8).
[tex]-\frac{8y}{-8}=\frac{4}{-8}-\frac{4x}{-8}[/tex][tex]y=-\frac{1}{2}+\frac{1}{2}x[/tex][tex]y=\frac{1}{2}x-\frac{1}{2}[/tex]Which is of the form
[tex]y=mx+b[/tex]where slope,m=1/2.
We know that the slope of the parallel lines is the same.
The slope of the required line is m =1/2.
Consider the line equation.
[tex]y=mx+b[/tex]Substitute x =1/9, y=-3, and m=1/2 in the equation to find the value of b.
[tex]-3=\frac{1}{9}(\frac{1}{2})+b[/tex][tex]-3=\frac{1}{18}+b[/tex]Subtract 1/18 from both sides.
[tex]-3-\frac{1}{18}=\frac{1}{18}+b-\frac{1}{18}[/tex][tex]-3\times\frac{18}{18}-\frac{1}{18}=b[/tex][tex]\frac{-54-1}{18}=b[/tex][tex]b=-\frac{55}{18}[/tex]Substitute m=1/2 and b =-55/18 in the line equation.
[tex]y=\frac{1}{2}x-\frac{55}{18}[/tex]Multiply both sides by 18.
[tex]18y=18\times\frac{1}{2}x-18\times\frac{55}{18}[/tex][tex]18y=19x-55[/tex]Final answer:
[tex]18y=19x-55[/tex]Given the dot product w•w = 29, find the magnitude of w.
Given the dot product expression as shown:
[tex]w\cdot w=29[/tex]Determine the value of 'w"
[tex]w^2=29[/tex]Take the square root of both sides to have:
[tex]\begin{gathered} \sqrt{w^2}=\pm\sqrt{29} \\ w=\pm\sqrt{29} \end{gathered}[/tex]Since we only need the magnitude of "w" and the magnitude is the positive value of the variable, hence;
[tex]|w|=\sqrt{29}[/tex]This gives the modulus of "w"
match each vertex in triangle EFG to it corresponding vertex in the dashed triangle
The matching is as following:
[tex]\begin{gathered} E\rightarrow H \\ F\rightarrow E \\ G\rightarrow G \end{gathered}[/tex]