Part (a)
S1 = 1/2 since we only have one term, i.e. we aren't really summing anything here.
S2 = (1/2) + (1/4) = (2/4) + (1/4) = (2+1)/4 = 3/4
S3 = (1/2) + (1/4) + (1/8) = (4/8) + (2/8) + (1/8) = (4+2+1)/8 = 7/8
Or,
S3 = S2 + 1/8 = (3/4) + (1/8) = (6/8) + (1/8) = (6+1)/8 = 7/8
S4 = S3 + 1/16 = (7/8) + (1/16) = (14/16) + (1/16) = (14+1)/16 = 15/16
S5 = S4 + 1/32 = (15/16) + (1/32) = (30/32) + (1/32) = (30+1)/32 = 31/32
Here's the summary:
S1 = 1/2S2 = 3/4S3 = 7/8S4 = 15/16S5 = 31/32=============================================================
Part (b)
Through trial and error, the formula is
Sn = ( (2^n) -1 )/(2^n)
For example, if n = 4, then,
Sn = ( (2^n) -1 )/(2^n)
S4 = ( (2^4) -1 )/(2^4)
S4 = (16 - 1)/(16)
S4 = 15/16
=============================================================
Part (c)
u1 = first term = 1/2
r = common ratio = 1/2
Sn = u1*( 1 - r^n)/(1 - r)
Sn = (1/2)*( 1 - (1/2)^n )/(1 - 1/2)
Sn = (1/2)*( 1 - (1/2)^n )/(1/2)
Sn = 1 - (1/2)^n
Sn = 1 - 1/(2^n)
Sn = (2^n)/(2^n) - 1/(2^n)
Sn = ( (2^n) -1 )/(2^n)
This helps confirm the formula mentioned in part b
=============================================================
Part (d)
As n gets very large, Sn approaches 1
Examples:
n = 7 leads to S7 = 0.9921875n = 10 leads to S10 = 0.9990234375n = 12 leads to S12 = 0.999755859375n = 20 leads to S20 = 0.999999046325684Try out larger values of n and the Sn value should get closer to 1.
The reason why this works is because Sn = (x - 1)/x where x = 2^n
The (x-1)/x is equivalent to 1 - (1/x) and the 1/x approaches zero when x gets larger. The x gets larger as 2^n does.
The value of Sn itself will never equal 1 exactly. It simply gets closer and closer.
=============================================================
Part (e)
The diagram is missing, so I don't have enough info to be able to answer this.
you're given a fair coin. you flip the coin until either heads heads tails (hht) or heads tails tails (htt) appears. is one more likely to appear first? if so, which one and with what probability?
The HHT is more likely to appear first with a probability of 2/3.
What is probability?The probability is an estimation of the likelihood that an event will occur. It assesses the event's certainty.
The formula for probability is given as;
P(E) = Number of Favourable Outcomes/Number of total outcomes.
Now according to the given question;
Only the second and third flips in the sequence matter. We understand the first flip is an H, and we're not interested in sequences that begin with T.
Lets start with the given format;
H _ _
If we get an HH_, the game is over and HHT will surely win (regardless of the number of heads we get, the next T is an automatic win). If we get an HTT, the game is over.
If we get an HTH, the game is a tie, and we must restart it.
Thus,
P(HHT) = 1/2
P(HTT) = 1/4
P(Draw) = 1/4
P(HHT wins) = P(HHT) / [1-P(Draw)]
= (1/2) / (1-1/4)
P(HHT wins) = 2/3
Thus, the probability of occurrence of HHT has more chances with 2/3.
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Multiply and simplify completely
√6 x 3√54
Answer:
54
Step-by-step explanation:
using the rule of radicals
[tex]\sqrt{ab}[/tex] = [tex]\sqrt{a}[/tex] × [tex]\sqrt{b}[/tex] and [tex]\sqrt{a}[/tex] × [tex]\sqrt{a}[/tex] = a , then
[tex]\sqrt{6}[/tex] × 3[tex]\sqrt{54}[/tex]
= [tex]\sqrt{6}[/tex] × 3 × [tex]\sqrt{9(6)}[/tex]
= [tex]\sqrt{6}[/tex] × 3 × [tex]\sqrt{9}[/tex] × [tex]\sqrt{6}[/tex]
= [tex]\sqrt{6}[/tex] × [tex]\sqrt{6}[/tex] × 3 × 3
= 6 × 9
= 54
[tex]{ \qquad\qquad\huge\underline{\boxed{\sf Answer}}} [/tex]
Here's the solution ~
[tex]\qquad \sf \dashrightarrow \: \sqrt{6} \times 3 \sqrt{54} [/tex]
[tex]\qquad \sf \dashrightarrow \: \sqrt{2 \times 3} \times \sqrt{2 \times {3}^{3} } [/tex]
[tex]\qquad \sf \dashrightarrow \: \sqrt{ {2}^{2} \times {3}^{4} } [/tex]
[tex]\qquad \sf \dashrightarrow \:2 \times {3}^{2} [/tex]
[tex]\qquad \sf \dashrightarrow \:2 \times 9[/tex]
[tex]\qquad \sf \dashrightarrow \:18[/tex]
a11
for -4, 8, -16, 32, …
are there any points with integer coordinates on a line if a line passes through (5,-8.5) and (13.8, 6.9)?
please provide a yes or no and examples.
There are at least two integer points in the line that passes through (5, - 8.5) and (13.8, 6.9).
Does a line contain coordinates with only integers?
In this problem we have a line that passes through two points on a Cartesian plane and herein we must check if that line contains at least one point with integer coordinates. First, determine the equation of the line:
Slope
m = [6.9 - (- 8.5)] / (13.8 - 5)
m = 7 / 4
Intercept
b = - 8.5 - (7 / 4) · 5
b = - 69 / 4
The equation of the line is y = (7 / 4) · x - 69 / 4.
Second, evaluate the equation of the line at different integers:
x= 6
y = (7 / 4) · 6 - 69 / 4
y = - 27 / 4
x = 7
y = (7 / 4) · 7 - 69 / 4
y = - 5
x = 8
y = (7 / 4) · 8 - 69 / 4
y = - 13 / 4
x = 9
y = (7 / 4) · 9 - 69 / 4
y = - 3 / 2
x = 10
y = (7 / 4) · 10 - 69 / 4
y = 1 / 4
x = 11
y = (7 / 4) · 11 - 69 / 4
y = 2
x = 12
y = (7 / 4) · 12 - 69 / 4
y = 15 / 4
x = 13
y = (7 / 4) · 13 - 69 / 4
y = 11 / 2
There are at least two integer points in the line that passes through (5, - 8.5) and (13.8, 6.9).
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please help im about to fail math solve for x: 5/2x+1/2x=7+7/2x
Answer:
yall smart
Step-by-step explanation:
find the value of x:
Answer:
The answer to the question is 35 degrees
What reasoning shows correctly how to rewrite 4 to the power of 3 over 2 end exponent?
The reasoning which shows the given statement correctly is [tex]4^{\frac{3}{2}}[/tex].
Given the statement is "4 to the power of 3 over 2 end exponent".
In mathematics, an expression is defined as a set of numbers, variables, and operations that are formed according to context-dependent rules.
In the given statement 3 over 2 means, 3 is divisible by 2 and it can be written as 3/2.
And it is also given that 4 to the power of 3 over 2 that means we have to write 3/2 in the power of 4 and it can be written as
[tex]4^{\frac{3}{2}}[/tex]
Hence, the reasoning which shows the given statement "4 to the power of 3 over 2" correctly is [tex]4^{\frac{3}{2}}[/tex].
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Show how 2/5 + 4/10 is the same as 4/5
Answer:
Explained through Step-by-step (Showed Work)
Step-by-step explanation:
Question: Show how 2/5 + 4/10 is the same as 4/5
Step-by-step:
2/5 and 4/10 are fractions. (So is 4/5)
To add fractions together, we need the denominators to be the same. This means that the bottom number of 2/5, which is 5 and the bottom number of 4/10 which is 10 have to be the same.
The easiest way to do this is to simplify 4/10. To simplify 4/10, we divide both the numerator (top number) and the denominator (bottom number) by the Greatest Common Factor.
In this case, 2 is the Greatest Common Factor because 4 divided by 2 is 2 and 10 divided by 2 is 5.
so, 4/10 can be simplified to 2/5.
Now, we can add 2/5 and 2/5 together by only adding the numerators. 2 + 2 = 4
so, 2/5 + 2/5 = 4/5
4/5 = 4/5
Therefore, 2/5 + 4/10 is the same as 4/5.
Hope this helps
solve the function please
for f(x) = x^2 - 2x + 1 , find f(x+h)-f(x) / h , h ≠ 0
For the given function f(x) = x^2 - 2x + 1, the difference quotient is {2x + h - 2]
How to find the difference quotient for the function f(x)?
Here we have the function f(x) = x^2 - 2x + 1
And we want to find the difference quotient, which is:
( f(x + h) - f(x))/h
Replacing the actual function there we get:
[ (x + h)^2 - 2*(x + h) + 1 - x^2 + 2x - 1]/h
Now we can simplify the numerator, we will get:
[ x^2 + 2xh + h^2 - 2x - 2h + 1 - x^2 + 2x -1]/h
[2xh + h^2 - 2h]/h
Taking the quotient we get:
{2x + h - 2]
That is the difference quotient for the function f(x).
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Write a linear function f(x)=mx+b for the table
The standard form the linear function is f(x)= -3x-3.
Linear FunctionAn equation can be represented by a linear function. The standard form for the linear equation is: y= mx+b , for example, y=5x+3. Where:
m= the slope. It can be calculated for Δy/Δx .
b= the constant term that represents the y-intercept.
For the given example: m=5 and b=3.
You should replace in the minimum two given values for x and y in the standard form for a linear equation mx+b . Thus, you will write a system linear where from it, you can find the coefficients m and b. See below.
For x=0 and y=-3y=mx+b
-3=0*m+b
-3=b
For x=-2 and y=3y=mx+b
3= -2*m+b
Now, you find b=-3, then
3= -2*m-3
3+3= -2m
6=-2m
m= -3
Thus, you can write f(x)= -3x-3.
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What value of "x" makes the equation -4x+7=2x+13 true?
OA. -4
OB. -1
OC. 2
OD. O
Answer:
B. -1
Step-by-step explanation:
Let's solve your equation step-by-step.
−4x+7=2x+13
Step 1: Subtract 2x from both sides.
−4x+7−2x=2x+13−2x
−6x+7=13
Step 2: Subtract 7 from both sides.
−6x+7−7=13−7
−6x=6
Step 3: Divide both sides by -6.
−6x/−6 = 6/−6
x=−1
Which table has a constant of proportionality between y and az of 0.6?
Choose 1 answer:
(G
z
Y
47
6
8
-8
4
9
14
H
10
13
y
2.4
5.4
8.4
y
3
2
9 6
15 10
Answer:
C
Step-by-step explanation:
if you divide the y by x, the answer it will give you will be 2/3 or .6
Solve 6-2x=3x
What would the answer be and step by step guide to completing it
Answer:
x=6/5
Step-by-step explanation:
6=2x+3x
5x=6
x=6/5
alysha is at a local library with her best friend. they've noticed a lot of cute boys in the library, and alysha bets that of the next twenty people to enter the library, exactly 12 will be cute boys. assuming there is a 50% chance that the next person to walk into the library is a cute boy, what is the probability that alysha will win the bet?
The probability that Alysha will win the bet is 0.12013
Let ‘x’ be the number of cute boys from next 20 persons that enter the library
And assume x be the integral values from 0 to 20
Given, the probability is 50% that the next person that enters the library is a cute boy,
total number of trials n=20 and probability of success p=50% = 0.5 = 1/2.
Hence, the p.m.f of X is given by
fX(x) = 20Cx(1/2)x(1−1/2)20−x
=20Cx(1/2)x(1/2)20−x
=20Cx(1/2)20
=20!(20−x)!x!⋅ 1220
=0,1,2,…,19,20
and the probability that X=x,
thus
P(X=x)=20!(20−x)!x!⋅1220
x=0,1,2,…,19,20
P(X=x)=20!(20−x)!x!⋅1220
x=0,1,2,…,19,20.
In particular, for X=x=12
P(X=12)=20!(20−12)!12!⋅1220
=20!8!12!⋅1220
≈0.12013
Hence, the probability that exactly 12 persons of the next 20 persons that walk into the library are cute boys is only approximately 0.12013.
Therefore, 0.1203 is less than 0.5, then it is more probable that Alysha will lose the bet than win it.
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You start at (6,4) and move 4 units left and 2 units down what point will you end up on?
Answer:(2,2)
Step-by-step explanation:
because if you follow the steps you will end up there
Sarah had a balance of $162 in her bank account at the start of the week. She withdrew $63.50 on Monday, $23.25 on Wednesday, and $26.45 on Thursday. On Friday, she deposited $87.30.At the end of the week, Sarah had ___(less or more) money in her savings account than she had at the beginning of the week.
How much more or less? $_______
Answer:
Less; $25.90 less than at the beginning of the week
Step-by-step explanation:
162-(63.50+23.25+26.45)=162-113.20=48.80
48.80+87.30=136.10; 162>136.10
162-136.10=Δ$25.90
$25.90 less
b. Parallel lines are lines that will never intersect. Do parallel lines exist in spherical geometry? Explain your answer.
Yes parallel line exist in spherical geometry. The radius or diameter and the tangent can be a pair of parallel line in spherical geometry
How parallel can exist in spherical geometryThe tangent of the sphere refers to a line that make contact with only a point in the sphere. The tangent is perpendicular to the center and the radius can be drawn to be parallel to the tangent.
The radius refers to a line running from the center to an end of the circle. Two radius can be drawn say radius A and radius B. Radius A will be drawn perpendicular to the tangent and radius B drawn perpendicular to radius A.
The pair of lines tangent and radius B will be parallel to each other.
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Describe a real-world situation that can be modeled by the equation y = 1/20x be sure to describe which each variable represents
The equation can be modelled to a real scenario where every purchase can be use to calculate sales tax.
How to find a real world situation that models an equation?The equation to be modelled is as follows:
y = 1 / 20x
let
x = The total price of purchases
y = The amount of sales tax
This simply means a person pays 1 / 20 or 5% of every purchase he/she makes.
The total price of the purchase(including tax) is x while the actual amount of tax paid is y.
Let's say you buy a goods worth 400 dollars. Let's calculated the tax as follows:
x = $400
Therefore,
y = 1 / 20 × 400
y = 1 / 20 × 400
y = $20
Hence, the tax for a purchase of 400 dollars is 20 dollars.
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What is the answer rounded to the
hundredths
place?
9.3478 m
+2.73 m
12.0778 m
➜>>
→ [?] m
Answer:12.8
Step-by-step explanation: u just round it
What is the answer to
|x+8|=|2x+1| ?
Answer:
[tex] \sf \large \: x=7 or x=−3[/tex]
Step-by-step explanation:
Let's solve your equation step-by-step.
|x+8|=|2x+1|
Solve Absolute Value.
|x+8|=|2x+1|
We know eitherx+8=2x+1orx+8=−(2x+1)
x+8=2x+1(Possibility 1)
x+8−2x=2x+1−2x(Subtract 2x from both sides)
−x+8=1
−x+8−8=1−8(Subtract 8 from both sides)
−x=−7
−x/−1 = −7/−1
(Divide both sides by -1)
x=7
x+8=−(2x+1)(Possibility 2)
x+8=−2x−1(Simplify both sides of the equation)
x+8+2x=−2x−1+2x(Add 2x to both sides)
3x+8=−1
3x+8−8=−1−8(Subtract 8 from both sides)
3x=−9
3x/3 = −9/3
(Divide both sides by 3)
x=−3
Check answers. (Plug them in to make sure they work.)
x=7(Works in original equation)
x=−3(Works in original equation)
Answer:
7
Step-by-step explanation:
x+8=2x+1
Subtract x on both sides. Subtract 1 on both sides
x=7
10. Sam earns a salary of $25,000 while Charlie earns a salary of $30,000. They each receive
a $2000 raise. Which of the following is true?
(a) Sam's raise is larger than Charlie's in both absolute and relative terms.
(b) The raises are the same in absolute terms, but Sam's is larger in relative terms.
(c) The raises are the same in relative terms, but Sam's is larger in absolute terms.
(d) The raises are the same in absolute terms, but Charlie's is larger in relative terms.
If the salary of Sam is $25000 and Charlie earns $30000 and each get a raise of $2000 then the statement "The raises are the same in absolute terms, but Charlie's is larger in relative terms" is true.
Salary of Sam = $25000
Raise in Sam's Salary = $2000
New Total Salary = $25000 + $2000 = $27000
Salary of Charlie = $30000
Raise in Charlie's Salary = $2000
New Total Salary = $30000 + $2000 = $32000
Hence the absolute value of their salaries are same but Charlie will get more in relative terms.
Hence, option d is correct.
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theo needs 1 1/2 cups of flour for one recipe and 2 3/4 cups of flour for a second recipe. Theo only has a 1/4 cup measuring cup. How many times will theo need to fill the 1/4 cup measuring cup to measure the flour he needs for the two recipes?
17 times theo will need to measure the flour for both the recipes.
WHAT IS BODMAS ?Children can use the acronym BODMAS to help them remember the right order to carry out mathematical operations while solving problems. B-Brackets, O-Orders (powers/exponents or roots), D-Division, M-Multiplication, A-Addition, and S-Subtraction are all abbreviated as BODMAS.
The order of operations (or operator precedence) in mathematics and computer programming refers to a set of rules that represent standards for which operations to carry out first in order to evaluate a specific mathematical expression. For instance, since the advent of contemporary algebraic notation, multiplication has been given a higher priority than addition in mathematics and most computer languages.
As a result, 1 + (2 3) = 7 rather than 1 + (2 3) = 9 is the value assigned to the expression 1 + 2 3. Exponents were first used in the 16th and 17th centuries, when addition and multiplication were given priority. Exponents could only be used as a superscript to the right of their base. Consequently, 3 + 52 = 28 and 3 52 = 75.
CALCULATIONfirst add the total amount of flour =
= [tex]1\frac{1}{2} + 2\frac{3}{4}[/tex]
= 1.5 + 2.75
= 4.25
no. of times = 4.25 * 4/1 = 17 times
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HELP PLEASEEE
If g (x) = x² + 6 and ƒ (x) = 2x − 4, find g (ƒ (x))
Step-by-step explanation:
Plug in 2x-4 in the function g(x)
[tex]g(f(x)) = {(2x - 4)}^{2} + 6 \\ \\ g(f(x)) = (2x - 4)(2x - 4) + 6 \\ = 4 {x}^{2} - 8x - 8x + 16 + 6 \\ g(f(x)) = 4 {x}^{2} - 16x + 22[/tex]
HOPE THIS HELPS
Jenna can't wait to play in her new sandbox! Her mom fills the sandbox with seven 50-pound bags of sand. Then, Jenna uses her bucket to dump out sand 10 times. If she dumps out 8 ounces of sand each time, how many pounds of sand are left in the sandbox?
Answer:
345 pounds
Step-by-step explanation:
Each bucket dumps out 8 ounces of sand
8 ounces = 1/2 pound
So total sand dumped 10 times = 1/2 x 10 = 5 pounds
Originally sandbox had 7 x 50 = 350 pounds of sand
Sand left in box = 350 - 5 = 345 pounds
A sum of £350 is invested at simple interest and amounts to £476 after 3 years. Calculate
(a) the total interest
(b) the rate per cent per annum.
Answer:
(a) $126
(b) .453333%
Step-by-step explanation:
(a) 476 - 350 = $126
(b)
[tex]si = p \times r\times t \\ 476 = 350 \times r\times 3 \\ 476 = 1050r \\ \frac{476}{1050} = r \\ r = 0.453333\%[/tex]
Help I have no clue what are the answers I tried 30 times!
The correct options for the linear combinations of the vectors are given by:
A, B, C, D, E and F.
What is a span between two vectors?The span between two vectors is the set that contains all linear combinations between these two vectors.
Supposing we have two vectors u1 and u2, as is the case in this problem, the infinitely many linear combinations have the following format:
k1u1 + k2u2
In which k1 and k2 are real numbers.
With k1 = 4 and k2 = -7, we get that option A is correct.
For option B, we have to see if it is possible to solve the following system.
k1 + 4k2 = 0.5k1 + 22k2 = 0.-4k1 - 11k2 = 0.The solution is k1 = k2 = 0, which are real numbers, hence option B is correct.
Both vectors u1 and u2 are part of the span, hence options D and F are correct. The size of the span is of infinity, hence option E is correct while option G is not.
The dimension of the span is of 3, as the vectors have 3 elements, hence option C is correct.
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What should I do when I finish High school and college
Answer: You should try to figure out the job you want and do that and maybe make a bucket list. Also, live your future out and your hapiness is what counts.
Step-by-step explanation:
HELP ME QUICK!!
Joe wants to wallpaper the back wall of his bedroom. The wall is in the shape of a rectangle its length of 17 feet and its width is 13 feet suppose wallpaper cost five dollars for each square foot how much will paper cost for the wall
Based on the shape of the back wall of Joe's bedroom and the dimensions of the rectangle, and the cost per square foot, the paper cost of the wall would be $1,105
What is the paper cost of the wall?First, find the area of the back wall of Joe's bedroom because this is the area that would need to be covered by the wall paper:
Area of rectangle = Length x Width
Solving gives:
= 17 x 13
= 221 square foot
The cost to cover the wall based on the cost in dollars of each square foot is:
= Number of square feet x Cost per square foot
= 221 x 5
= $1,105
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Please help me with this geometry question. Answer should be entered like this (answer, answer) thank you
Answer:
Q is the point (3, 4).
Step-by-step explanation:
The formulae for the coordinates of a point (h, k) which divides a line joining the points (x1, y1) and (x2, y2) in the ratio m : n is:
h = (mx2 + nx1) / (m + n), k = (my2 + my1)/(m+n)
Now we have S = (x1, y1) = (-2, -6) and T = (x2, y2) = (5, 8) and
m = 5 and n = 2.
So here, for point Q:
h = (5*5) + 2*-2)/(2+5)
= 21/7
= 3.
k = (5*8 + (-6*2) / 7
= 28/7
= 4.
if f(x) = x and g(x) = 2, what is (f.g) (x)?
The value of (f.g) (x) is 2. Option B
What is a function?A function can be defined as an expression or rule that explains the relationship between an independent and dependent variable.
Given the functions;
f(x) = xg(x) = 2To determine (f.g) (x), we need to know that it is a composite function
We substitute the value of g(x) = 2 as x in f(x)
But we have f(x) = x
Then,
(f.g) (x) = 2
Thus, the value of (f.g) (x) is 2. Option B
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