Let [tex]S_n[/tex] be the [tex]n[/tex]-th term in this sequence, so
[tex]S_1 = 1[/tex]
[tex]S_2 = 1 + 2[/tex]
[tex]S_3 = 1 + 2 + 2^2[/tex]
and so on, with general [tex]n[/tex]-th term
[tex]S_n = 1 + 2 + 2^2 + \cdots + 2^{n-1} = \displaystyle \sum_{i=1}^n 2^{i-1}[/tex]
Observe that
[tex]2 S_n = 2 + 2^2 + 2^3 + \cdots + 2^n[/tex]
and by subtracting this from [tex]S_n[/tex], we eliminate all but the outermost terms, namely
[tex]S_n - 2S_n = 1 - 2^n[/tex]
so that
[tex]-S_n = 1 - 2^n[/tex]
[tex]S_n = 2^n - 1[/tex]
Now, the sum of the first [tex]n[/tex] terms of the sequence [tex]\{S_1,S_2,S_3,\ldots,S_n\}[/tex] is
[tex]S = S_1 + S_2 + S_3 + \cdots + S_n[/tex]
[tex]S = \displaystyle \sum_{i=1}^n S_i[/tex]
[tex]S = \displaystyle \sum_{i=1}^n (2^i - 1)[/tex]
[tex]S = \displaystyle \sum_{i=1}^n 2^i - \sum_{i=1}^n 1[/tex]
[tex]S = \displaystyle 2 \underbrace{\sum_{i=1}^n 2^{i-1}}_{S_n} - n[/tex]
[tex]S = 2(2^n - 1) - n[/tex]
[tex]S = \boxed{2^{n+1} - n - 2}[/tex]
what is the binomials of x^2+5x-36
Answer:
(x+9) and (x-4)
Step-by-step explanation:
Jennifer went to a farm and bought tomatoes
for $2 per pound and garlic for $3 per pound.
If Jennifer spent $72 on tomatoes and garlic,
how many pounds can she buy if she buys
only garlic?
Answer:
24 pounds of garlic
Step-by-step explanation:
if she spends 3$ per pound, you divide 72 by 3. the answer to that is 24, so if she spends 72 dollars on only garlic, she can buy 24 pounds.
Jennifer can buy 24 pounds of garlic
we see here, that if she spends 3$ per pound, you divide 72 by 3.
The answer would be 24, so if she spends 72 dollars on only garlic,
she can buy 24 pounds.
What is a pound defined as?1: any of various units of mass and weight specifically: a unit now in general use among English-speaking peoples equal to 16 avoirdupois ounces or 7000 grains or 0.4536 kilogram — see Weights and Measures Table
2: the basic monetary unit of the United Kingdom.
What is a division example?In maths, a division is a process of splitting a specific amount into equal parts. For example, we can divide a group of 20 members into 4 groups of 5 members each or 5 groups of 4 members each
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The graph shows the volume of the ice cubes versus temperatures in degrees Celsius. Which statement is false
The false statement include:
Volume is explanatory variables.Data shows a positive linear relationship.What is a Positive linear relationship?
This relationship between two variables involves them having a positive slope. This means that as volume increases, temperature also increases which isn't the case.
This cause and effect pattern also means that the volume doesn't have explanatory variables.
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Answer:
Volume is the explanatory variable
Step-by-step explanation:
because vanilla is better than chocolate
Use the difference between -2 and 5 to find the distance. The distance is
Answer:
It must be probably √29
Step-by-step explanation:
[tex]distance = \sqrt{ { (- 4)}^{2} + {(5)}^{2} } [/tex]
hope it to be the correct...
11. Find the area of the given figure?
Answer:
9.62 cm²
Step-by-step explanation:
Area of sectordiameter = 7 cm
r = 7 ÷2 = 3.5 cm
Ф = 45°
[tex]\sf \boxed{\text{\bf Area of sector = $\dfrac{\theta}{360}*\pi r^2$}}[/tex]
[tex]\sf =\dfrac{45}{360}*3.14*3.5*3.5\\\\ = 4.81 \ cm^2[/tex]
Area of the given sectors = 2 *4.81
= 9.62 cm²
Hello!
Let's solve this in step
Let's find the circular region to the left of the center:
[tex]Area = \frac{45}{360}*Area_.formuka_.of_.Circle =\\ Area = \frac{45}{360}*\pi (radius)^2\\ Area = \frac{45}{360} *\pi (3.5)^2\\Area=4.81[/tex]
Now let's find the circular region to the right of the center
⇒ the angle 45° by the vertical angle theorem is equal to the angle
opposite it in the other circular region
⇒ has the same area as the left region
Total Area = 4.81 + 4.81 = 9.62 cm²
Answer: 9.62 square centimeters
Hopefully that helps!
What is the simplified form of i^15 ?
what type of polynomial is -3
Answer:
a monomial, a constant
Step-by-step explanation:
It has only 1 term, so it's a monomial.
Also, the term is a constant.
Which graph correctly shows f(x)=x+3 / x^2-4
Step-by-step explanation:
it isnot looking well here so go on geogebra. org/calculator and you will see. have a nice day :))
Answer:
C on Edge 2022
Step-by-step explanation:
Just did it and got it right
An airplane travels 640 miles from Topeka to Houston in 3.2 hours, going against the wind. The return trip is with the wind, and takes only 2 hours. Find the rate of the airplane with no wind. Find the rate of the wind.
A) The airplane flies at 210 mi/h with no wind. The rate of the wind is 50 mi/h.
B) The airplane flies at 210 mi/h with no wind. The rate of the wind is 60 mi/h.
C)The airplane flies at 260 mi/h with no wind. The rate of the wind is 60 mi/h.
D)The airplane flies at 260 mi/h with no wind. The rate of the wind is 50 mi/h.
The airplane flies at 260 mi/h with no wind, while the rate of the wind is 60 mi/h.
What is an equation?An equation is an expression that shows the relationship between two or more numbers and variables.
Let a represent the rate of the airplane with no wind. and b the rate of the wind, hence:
(a - b)3.2 = 640
a - b = 200 (1)
Also:
(a + b)2 = 640
a + b = 320 (2)
The solution to equation 1 and 2 is:
a = 260, b = 60
The airplane flies at 260 mi/h with no wind, while the rate of the wind is 60 mi/h.
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Work out
(4 × 10³) – (7 × 10²)
Give your answer in standard for
Answer:
3300
Step-by-step explanation:
This can be worked out using PEMDAS:
First you solve in the PARENTHESIS:
(4 × 10³) – (7 × 10²)
Then, you look at the EXPONENTS:
(4 × 10³) = (4 × 1000)
(7 × 10²) = (7 × 100)
After that, look at the MULTIPLICATION:
(4000) - (700)
Since there is only subtraction left, subtract it:
3300
Answer:
Parenthesis:
4 *1000 = 4000
7*100= 700
4000-700=3300
Step-by-step explanation:
2. Try It #2 The population of a small town increased from 1,442 to 1,868 between 2009 and 2012 . Find the change of population per year if we assume the change was constant from 2009 to 2012 .
Answer:
If it is assumed that the change was constant from 2009 and 2012 , then change of population is 142 .
Step-by-step explanation:
In the question, it is given that the population of a small town increased from 1442 to 1868 between 2009 and 2012 .
It is asked to find the change of population if it is assumed that the change was constant from 2009 and 2012 .
Step 1 of 2
Population in 2012 is given to be 1868 and
Population in 2009 is given to be 1442 .
Duration of year is given by 2012-2009 which is 3 years.
Change in population is given by 1868-1442 which is 426 .
Step 2 of 2
Change in population per year is given by dividing the change in population and the duration of years it took to happen the change.
Hence, change in population per year is
[tex]$\frac{426}{3}=142$[/tex]
Hence, the population was changed by 142 per year from 2009 to 2012 .
Helppppp !!!!
Which of the graphs below shows the solution set of [2,7]
Answer:
B
Step-by-step explanation:
[] means the endpoints are inclusive. Therefore [2,7] is equivalent to 2 <= x <= 7. So [2,7] represents graph B.
"I am thinking of a number. If you
multiply my number by 4 and then add 3
times my number, you will get 175. What is
my number?"
Answer:
y =25Step-by-step explanation:
y×4+3×y=175 4y+3y=175 7y=175 divide by 7 both side with make value of y y=25. this is my answerWhich expression is equivalent to the given expression?
Answer:
The third option.
Step-by-step explanation:
[tex](\frac{4xy^3}{x^2})^2\\\\=(\frac{4y^3}{x})^2\\\\=\frac{16y^6}{x^2}[/tex]
Hence the 3rd option.
Celo used the simple interest formula uppercase a = uppercase p (1 r t) to calculate the interest he earned on his savings last month. which equation is equivalent to the simple interest formula?
The equation which is equivalent to the given simple interest formula is
[tex]P=\frac{A}{1+rt}[/tex].
What is an equivalent equation?
Equivalent equations are algebraic equations that have identical solutions.
According to the given question.
We have a simple interest formula.
[tex]A=P(1+rt)[/tex]
Now, the equivalent equation for the above formula can be given as
[tex]A=P(1+rt)[/tex]
⇒[tex]P=\frac{A}{1+rt}[/tex]
Hence, the equation which is equivalent to the given simple interest formula is [tex]P=\frac{A}{1+rt}[/tex].
Thus, option A is correct.
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Which graph represents an exponential function?
The first graph is the exponential function.
What is exponential function?The exponential function serves as the positive-valued function with respect to real variable. and the values denoted as X which is exponential is all real numbers.
Analyzing the first graph, moving down the graph, we can see that x value increases.
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Answer:
First Graph
Step-by-step explanation:
Edge
Helppp meeeeee pleasseeeeeeeeeee
Answer:
Khan Academy.
Step-by-step explanation:
You can go on khan academy and search up "percentages and numbers." The videos should immediately pop up. there are also lessons if you don't want to risk getting your problem wrong. If you are still confused and do not understand please just comment and I will respond.
How many units are in the sum of the lengths of the three altitudes in a triangle with sides $7,$ $24,$ and $25$
56 units are in the sum of the lengths of the three altitudes in a triangle with sides $7,$ $24,$ and $25$
Properties of triangles are:
A triangle has three sides and three angles.
The sum of the angles of a triangle is always 180 degrees.
The exterior angles of a triangle always add up to 360 degrees.
The sum of consecutive interior and exterior angle is supplementary.
How to solve?Given:
Height: ha = 7
Height: hb = 24
Height: hc = 25
Sum to sides is:
25 +7 + 24 =56 units.
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PLEASE HELP ASAP THANK YOU!!
Answer:
Second graph
Step-by-step explanation:
See attached image.
HELPPPPPPP PLEASEEEEEEE
Find the solutions of the quadratic equation:
Answer:
Answer A is correct
Step-by-step explanation:
Let us use completing square method for this.
[tex]x^{2} -5x -10=0\\\\x^{2} -5x=10\\\\x^{2} -5x+\frac{25}{4}=10+ \frac{25}{4}\\\\ (x+\frac{5}{2}) ^{2} =\frac{10*4}{1*4} +\frac{25}{4}\\\\ (x+\frac{5}{2}) ^{2} =\frac{40}{4} +\frac{25}{4}\\\\ (x+\frac{5}{2}) ^{2} =\frac{40+25}{4} \\\\(x+\frac{5}{2}) ^{2} =\frac{65}{4} \\\\(x+\frac{5}{2}) = \±\sqrt{\frac{65}{2} }\\\\(x+\frac{5}{2}) =\±\frac{\sqrt{65}}{4}\\\\x=\±\frac{\sqrt{65}}{4}-\frac{5}{2}[/tex]
∴ [tex]x=-\frac{5}{2} \±\frac{\sqrt{65}}{4}[/tex]
A company has 8 male and 6 female employees, and needs to nominate 2 men and 2 women for the company bowling team. How many different teams can be formed
The team can be formed in 420 different ways
This is a combination problem since we are to select a set of people from a group. Combination has to do with selection.
if r number of objects is to be selected from a pool of n objects, this can be done in nCr number of ways.
[tex]nCr=\frac{n!}{r!(n-r)!}[/tex]
Now If A company has 8 male and 6 female employees, and needs to nominate 2 men and 2 women for the company bowling team, then this can be done in the following way;
[tex]_{8} C_{2}=\frac{8!}{2!6!} =28\\_{6} C_{2}=\frac{6!}{2!4!} =15[/tex]
[tex]_{8} C_{2}*_{6} C_{2}=28*15[/tex]
[tex]=420[/tex] teams
Hence, The team can be formed in 420 different ways
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For the arithmetic sequence beginning with the terms {11, 17, 23, 29, 35 ...}, what is the sum of the first 17 terms?
a.)
1003
b.)
1116
c.)
896
d.)
1235
Answer:
a.) 1003
Step-by-step explanation:
The formula to find the sum of an arithmetic sequence: Sn = n/2[2a + (n - 1)d], where a is the first term, n is the number of terms and d is the common difference.
1) Let's substitute the values into the formula.
S17 = 17/2[2(11) + (17 - 1)6]
S17 = 17/2[22 + (16)6]
S17 = 17/2[22 + 96]
S17 = 17/2[118]
S17 = 17/2 × 118
S17 = 1003
Therefore, the sum of the first 17 terms of this arithmetic sequence is 1003 (a).
Question 17 of 25
how many solutions does the following system of equations have?
y=5/2x+2
2y=5x+4
a. zero
b. one
c.infinitely many
d. two
Answer:
c
Step-by-step explanation:
y = [tex]\frac{5}{2}[/tex] x + 2 → (1)
2y = 5x + 4 ( divide through by 2 )
y = [tex]\frac{5}{2}[/tex] x + 2 → (2)
the 2 equations are the same and will have infinitely many solutions
what is the positive angle less than 2π that is coterminal with 14π/6
Coterminal angles are angles that are made by the positive x-axis of the coordinate plane. The positive angle that is less than 2π and is coterminal with 14π/6 is 1/3π.
What are coterminal angles?Coterminal angles are angles that are made by the positive x-axis of the coordinate plane.
The angle that 14π/6 makes from the positive x-axis is 1/3π, which is equal to 60°.
Hence, the positive angle that is less than 2π and is coterminal with 14π/6 is 1/3π.
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the missing side of the triangle
Answer:
x = √95Step-by-step explanation:
from the sign on the picture it is a rigth triangle
we use pythagoras
x² = 12² - 7²
x² = 144 - 49
x² = 95
x = √95
(x = 9.75)
Answer:
x = [tex]\sqrt{95}[/tex]
Step-by-step explanation:
In this problem, we need to use the Pythagorean theorem.
Pythagorean theorem is a theorem where you can find a side length in a right triangle.
The hypotenuse (The longest side) squared equals the other two side lengths added together after being squared.
In numbers, this will be: (12)^2 = (7)^2 + (x)^2.
144 = 49 + x^2.
95 = x^2
x = [tex]\sqrt{95}[/tex]
We cannot simplify [tex]\sqrt{95}[/tex], so that is the final answer.
31. Lefties Find the sample size needed to estimate the percentage of California residents who are left-handed. Use a margin of error of three percentage points, and use a confidence level of 99%. a. Assume that pn and qn are unknown. b. Assume that based on prior studies, about 10% of Californians are left-handed. c. How do the results from parts (a) and (b) change if the entire United States is used instead of California
a. The required sample size when pn and qn are unknown then n = 1844
b. The required sample size when about 10% of Californians are left-handed is n = 664
c. There would have been no effect on parts a and b if entire US has been chosen
According to statement
a. No estimate of proportion is given so we will assume:
{p} = {q} = 0.5
For 99% confidence, z = 2.576
E = 0.03
Hence, Required sample size
n = (0.5)*(0.5)*(2.576/0.03)^2
n = 1844
b. {p} = 0.10
{q} = 1 - {p}
{q} = 1 - 0.10
{q} = 0.90
For 99% confidence, z = 2.576
E = 0.03
Hence, Required sample size n = (0.1)*(0.9)*(2.576/0.03)^2 = 664
c. There would have been no effect on parts a and b if entire US has been chosen because minimum sample size required is not dependent on population size.
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From the diagram below, given the side lengths marked, and if we know that < C is congruent to < E, we can say that ___.
Select one:
a.
the two triangles are similar by SAS
b.
the two triangles are not similar
c.
the two triangles are congruent
d.
the two triangles are similar by AA
Since we know that <C is congruent to <E, we can say that: A. the two triangles are similar by SAS.
The properties of similar triangles.In Geometry, two triangles are similar when the ratio of their corresponding sides are equal in magnitude and their corresponding angles are congruent.
Ratio = BC/AC = 6/3 = 2.
Ratio = FE/DE = 4/2 = 2.
Thus, the ratio of their corresponding sides are equal in magnitude i.e BC/AC ≡ FE/DE.
Since we know that <C is congruent to <E, we can say that the two triangles are similar by side, angle, side (SAS) criterion.
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KM=
Help me asap pleasee
Answer: 3
Step-by-step explanation:
By the intersecting chords theorem,
[tex](KM)(16)=(4)(12)\\\\KM=3[/tex]
Given that polygon PQRS≅ polygon LMNO, identify the congruent corresponding part for PQ
(NO GRAPH PROVIDED)
Answer:
LM
Step-by-step explanation:
Usually these problems would already be in order, so since PQ are the first 2 letters of polygon PQRS, then for LMNO, it's LM.
Find dy/dx by implicit differentiation.
(sin pix + cos piy)^8 = 17
dy/dx =
please help quickly
dy/dx by implicit differentiation is cos(πx)/sin(πy)
How to find dy/dx by implicit differentiation?Since we have the equation
(sin(πx) + cos(πy)⁸ = 17, to find dy/dx, we differentiate implicitly.
So, [(sin(πx) + cos(πy)⁸ = 17]
d[(sin(πx) + cos(πy)⁸]/dx = d17/dx
d[(sin(πx) + cos(πy)⁸]/dx = 0
Let sin(πx) + cos(πy) = u
So, du⁸/dx = 0
du⁸/du × du/dx = 0
Since,
du⁸/du = 8u⁷ and du/dx = d[sin(πx) + cos(πy)]/dx= dsin(πx)/dx + dcos(πy)/dx
= dsin(πx)/dx + (dcos(πy)/dy × dy/dx)
= πcos(πx) - πsin(πy) × dy/dx
So, du⁸/dx = 0
du⁸/du × du/dx = 0
8u⁷ × [ πcos(πx) - πsin(πy) × dy/dx] = 0
8[(sin(πx) + cos(πy)]⁷ × (πcos(πx) - πsin(πy) × dy/dx) = 0
Since 8[(sin(πx) + cos(πy)]⁷ ≠ 0
(πcos(πx) - πsin(πy) × dy/dx) = 0
πcos(πx) = πsin(πy) × dy/dx
dy/dx = πcos(πx)/πsin(πy)
dy/dx = cos(πx)/sin(πy)
So, dy/dx by implicit differentiation is cos(πx)/sin(πy)
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