Since [tex]a_1,a_2,a_3,\cdots[/tex] are in arithmetic progression,
[tex]a_2 = a_1 + 2[/tex]
[tex]a_3 = a_2 + 2 = a_1 + 2\cdot2[/tex]
[tex]a_4 = a_3+2 = a_1+3\cdot2[/tex]
[tex]\cdots \implies a_n = a_1 + 2(n-1)[/tex]
and since [tex]b_1,b_2,b_3,\cdots[/tex] are in geometric progression,
[tex]b_2 = 2b_1[/tex]
[tex]b_3=2b_2 = 2^2 b_1[/tex]
[tex]b_4=2b_3=2^3b_1[/tex]
[tex]\cdots\implies b_n=2^{n-1}b_1[/tex]
Recall that
[tex]\displaystyle \sum_{k=1}^n 1 = \underbrace{1+1+1+\cdots+1}_{n\,\rm times} = n[/tex]
[tex]\displaystyle \sum_{k=1}^n k = 1 + 2 + 3 + \cdots + n = \frac{n(n+1)}2[/tex]
It follows that
[tex]a_1 + a_2 + \cdots + a_n = \displaystyle \sum_{k=1}^n (a_1 + 2(k-1)) \\\\ ~~~~~~~~ = a_1 \sum_{k=1}^n 1 + 2 \sum_{k=1}^n (k-1) \\\\ ~~~~~~~~ = a_1 n + n(n-1)[/tex]
so the left side is
[tex]2(a_1+a_2+\cdots+a_n) = 2c n + 2n(n-1) = 2n^2 + 2(c-1)n[/tex]
Also recall that
[tex]\displaystyle \sum_{k=1}^n ar^{k-1} = \frac{a(1-r^n)}{1-r}[/tex]
so that the right side is
[tex]b_1 + b_2 + \cdots + b_n = \displaystyle \sum_{k=1}^n 2^{k-1}b_1 = c(2^n-1)[/tex]
Solve for [tex]c[/tex].
[tex]2n^2 + 2(c-1)n = c(2^n-1) \implies c = \dfrac{2n^2 - 2n}{2^n - 2n - 1} = \dfrac{2n(n-1)}{2^n - 2n - 1}[/tex]
Now, the numerator increases more slowly than the denominator, since
[tex]\dfrac{d}{dn}(2n(n-1)) = 4n - 2[/tex]
[tex]\dfrac{d}{dn} (2^n-2n-1) = \ln(2)\cdot2^n - 2[/tex]
and for [tex]n\ge5[/tex],
[tex]2^n > \dfrac4{\ln(2)} n \implies \ln(2)\cdot2^n - 2 > 4n - 2[/tex]
This means we only need to check if the claim is true for any [tex]n\in\{1,2,3,4\}[/tex].
[tex]n=1[/tex] doesn't work, since that makes [tex]c=0[/tex].
If [tex]n=2[/tex], then
[tex]c = \dfrac{4}{2^2 - 4 - 1} = \dfrac4{-1} = -4 < 0[/tex]
If [tex]n=3[/tex], then
[tex]c = \dfrac{12}{2^3 - 6 - 1} = 12[/tex]
If [tex]n=4[/tex], then
[tex]c = \dfrac{24}{2^4 - 8 - 1} = \dfrac{24}7 \not\in\Bbb N[/tex]
There is only one value for which the claim is true, [tex]c=12[/tex].
The diagonal is a of a rectangular field is 169m. If the ratio of the length to the width is 12:5 find the dimensions
Based on the calculations, the dimensions of the rectangle are
Length, L = 156 meters.Width, w = 65 meters.What is a diagonal?The diagonal of a rectangle can be defined as a line segment that connects any two (2) of its non-adjacent vertices together while dividing the rectangle into two (2) equal parts.
Mathematically, the length of diagonals of a rectangle can be calculated by using this formula:
d = √(l² + w²)
Where:
d is the diagonal of a rectangle.l is the length of a rectangle.w is the width of a rectangle.Since the ratio of the length to the width is 12:5, we have:
Width, w = 5l/12
Substituting the given parameters into the formula, we have;
169 = √(l² + (5l/12)²)
169² = l² + (5l/12)²
169² = l² + (25l²/144)
Length, L = 156 meters.
For the width, we have:
Width, w = 5l/12
Width, w = 5(156)/12
Width, w = 65 meters.
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How many ways are there to get from (0, 0) to (7, 7) in the coordinate plane
with movements of only one unit right or one unit up? How many ways are there to do so that do not go above the line y = x?
The number of ways there are to move from (0, 0) to (7, 7) in the coordinate plane with movements of only one unit right or one unit up accordingly is; 49 while that such that y =x is; 7.
How many ways are there to get from (0, 0) to (7, 7) in the coordinate plane with pmovements of only one unit right or one unit up?It follows from the task content that the movement intended on the coordinate plane is; from (0, 0) to (7, 7).
The number of ways to move such that movements of only one unit right or one unit up is; 7 × 7 = 49.
The number of ways for which y= x is therefore is; 7 as the movement is diagonal.
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Question 5(Multiple Choice Worth 2 points)
(07.01 MC)
What is the range of the function f(x) = x - 3?
Answer:
(-∞, ∞)
Step-by-step explanation:
Unless there are restrictions on the domain, the range of any odd-degree polynomial function is "all real numbers."
Linear functionThe given function is a linear function, degree 1. This is an odd degree, so the range of the function is "all real numbers."
-∞ < f(x) < ∞
(-∞, ∞) . . . . . in interval notation
Jack and Serena can write a lab report together in 12 hours. If all people work at the same rate, how many hours would it take to write a lab report if Kay joins Jack and Serena?
O Yes
O No
2. (02.01 LC)
Is the following relation a function? (1 point)
{(3,-2), (1, 2), (-1,-4), (-1, 2)}
Answer:
no
Step-by-step explanation:
In a function, each "x" value [input] must have only one corresponding "y" value.
We know that points are written as (x, y)
Our points: {(3,-2), (1, 2), (-1,-4), (-1, 2)}
We can see that the x-input of -1 has two y-outputs, so this is therefore not a function
hope this helps! have a lovely day :)
11/6 + -2/5 +-13/10?
Answer:
2/15Step-by-step explanation:
11/6 + -2/5 +-13/10?
11/6 + (-2/5) + (-13/10)
43/30 - 13/10
2/15
How is a circumference related to an arc
Answer:
An arc of a circumference or of a circle mostly is a portion of the circumference. The length of an arc for all intents and purposes is simply the length of this portion of the circumference in a definitely major way. The circumference itself can particularly be considered an arc that goes around the circle in a fairly major way.
The marked price of a mobile set is Rs 9,600 and 40% discount is allowed to make 20% profit. By what percent is the discount to be reduced to increase the profit by 10%?
Answer:
5%
Step-by-step explanation:
price after discount :
9 600 - 9 600×40%
= 5760
Original price (price without profit) :
let x be the original price of the device.
x + x × 20% = 5760
Then
x = (5 760×100)÷120
= 4 800
Original price increased by 30% :
4 800 + 4 800×30%
= 6 240
the discount needed to increase the profit by 10% :
[(9 600-6 240)÷9 600]×100
= 35%
Then
to increase the profit by 10% ,we have to reduce
the percent of discount to :
40% - 35%
= 5%
What are the solutions to the system of equations? {−x+y=4y+12=x2+x
Answer: (-4, 0) and (4, 8)
Step-by-step explanation:
Looking at the graph, we can see where the line passes the parabola and can tell the solutions are (-4, 0) and (4, 8)
Factorise 10xy-12+15x-8y
Answer: 10xy-12+15x-8y=(5x-4)*(2y+3).
Step-by-step explanation:
[tex]10xy-12+15x-8y=(10xy-8y)+(15x-12)=\\=2y*(5x-4)+3*(5x-4)=(5x-4)*(2y+3).[/tex]
find the length of MN
Answer:
The length of MN is 5 cm
Step-by-step explanation:
Add the segment LX, parallel to QP.
Recall the properties of midsegment:
Midsegment is parallel to side,Midsegment is half the length of the parallel side.We have:
Since QL = LR, the point L is midpoint of Q,Since PN = NL, the point N is midpoint of PL,Since LX is parallel to QP, LX is midsegment of ΔPRQ.Find the length of LX:
LX = QP/2 = 20/2 = 10 cmSince QP ║ LX ║ NM, the segment NM is the midsegment of ΔPLX.
Find the length of NM:
NM = LX/2 = 10/2 = 5 cmOA=
Please help asap!! Thanks so much :))
In the given diagram, the value of the dashed side of rhombus OABC is 5
Distance between two pointsFrom the question, we are to determine the length of the dashed line (OA), in rhombus OABC
In the diagram, we can observe that the length of OA is the distance between point A and the origin (O).
Using the formula for calculating distance between two points,
d =√[(x₂-x₁)² + (y₂-y₁)²]
In the diagram,
The coordinate of the origin is (0, 0)
The coordinate of point A is (3, 4)
Thus,
x₁ = 0
x₂ = 3
y₁ = 0
y₂ = 4
Putting the parameters into the formula, we get
OA =√[(3-0)² + (4-0)²]
OA =√(3² + 4²)
OA =√(9+16)
∴ OA =√25
OA = 5
Hence, in the given diagram, the value of the dashed side of rhombus OABC is 5
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For a pair of sample x- and y-values, the ______________ is the difference between the observed sample value of y and the y-value that is predicted by using the regression equation.
For a pair of sample x and y values, the residual exists as the distinction between the observed sample value of y and the y value that exists indicated by utilizing the regression equation.
What is the difference between the observed sample value of Y and the Y value that exists as predicted by utilizing the regression equation?In regression analysis, the distinction between the observed value of the dependent variable and the predicted value exists named the residual.
A regression equation exists utilized in stats to estimate what relationship if any, exists between sets of data.
For a pair of sample x and y values, the residual exists as the distinction between the observed sample value of y and the y value that exists indicated by utilizing the regression equation.
Residual = observed y - predicted y.
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If ₹405 is to be divided among three persons A, B, C in the ratio of 3:5:7, how much money does each one get? Express them in percentages.
I will mark the first answerer as Brainliest.
Answer:
A = 20%
B = 33.33%
C = 46.67%
Step-by-step explanation:
Ok we need to add up 3, 5, and 7
3+5+7 = 15
405/15 = 27.
27*3 = A
27*5 = B
27*7 = C
A = 81 / 20%
B = 135 / 33.33%
C = 189 / 46.67%
Question 9 of 25 a bookstore costs $105 a day to keep open, and it spends $12 for each book that it sells. if each book sells for $15, what is the break-even point for this bookstore?
The store will break even after selling 35 books.
What is the break-even point?The break-even point is the moment at which total cost and total revenue are equal, implying that your small firm has no loss or benefit. In other words, you've reached the point where the costs of production equal the revenues for a product.To find what is the break-even point for this bookstore:
Given:
The cost required per day to keep the store open = $105Amount spent on each book = $12Let, n be the number of books.
Total costs = 12n + 105
The selling price of each book = $15
Total revenue = 15n
For breaking even, costs = revenue.
12n + 105 = 15n105 = 15n - 12n105 = 3n3n = 105Dividing both sides by 3:3n/3 = 105/3n = 35Therefore, the store will break even after selling 35 books.
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What is the solution to 3/2b + 5 < 17? Explain How.
(1) b < 8
(2) b > 8
(3) b < 18
(3) b > 18
We subtract 5 from both sides of the inequality.
3/2b + 5 - 5 < 17 - 53/2 b < 12Multiply both sides by 2/3.
( 2/3) * (3/2b) < (2/3) * 12b < 8Therefore, the correct option is alternative "A".
We would think that it is option B, but the only difference is that it changes the direction of the sign.
Answer: [A]: " b < 8 " .
_____
Step-by-step explanation:
Given:
Find the solution to: " 3/2b + 5 < 17 " ; and choose from the answer choices.
So; we have:
(3/2)b + 5 < 17 ;
Now, subtract "5" from each side of this inequality:
(3/2)b + 5 − 5 < 17 − 5 ;
To get:
(3/2)b < 12 ;
Now, let's multiply Each Side of this inequality by "2" ;
to get rid of the fraction:
" 2*(3/2)b < 12*2 " ;
{Note: " [tex]2 *\frac{3}{2}=\frac{2}{1}*\frac{3}{2}[/tex] " } ;
Note: To simplify: " [tex]\frac{2}{1} * \frac{3}{2}[/tex] " ;
Note the "2" in the denominator in the "first term" ;
And: The "2" in the denominator in the "second term" ;
Both "cancel out" to "1" ; since: "[ 2 / 2 = 2÷2 = 1 ]" ;
And: we have: " [tex]\frac{1}{1}*\frac{3}{1}= 1 *3 = 3[/tex] " };
_____
and rewrite:
" 3b < 24 " ;
Now, divide Each side of the inequality by "3" ;
to isolate "b" on one side of the inequality;
and to solve for "x" ;
_____
" 3b/3 < 24/3 " ;
to get:
_____
" b < 8 " ; which corresponds to the correct answer:
Answer choice: [A]: " b < 8 " .
_____
Hope this is helpful to you! Best wishes!
_____
Find the terms through degree four of the maclaurin series for f(x) = sin(x) 1−x.
The terms through degree four of the Maclaurin series is [tex]f(x)=x+x^{2} +\frac{5x^{3} }{6} +\frac{5x^{4} }{6} +.....[/tex].
In this question,
The function is f(x) = [tex]\frac{sin(x)}{1-x}[/tex]
The general form of Maclaurin series is
[tex]\sum \limits^\infty_{k:0} \frac{f^{k}(0) }{k!}(x-0)^{k} = f(0)+\frac{f'(0)}{1!}x+\frac{f''(0)}{2!}x^{2} +\frac{f'''(0)}{3!}x^{3}+......[/tex]
To find the Maclaurin series, let us split the terms as
[tex]f(x)=sin(x)(\frac{1}{1-x} )[/tex] ------- (1)
Now, consider f(x) = sin(x)
Then, the derivatives of f(x) with respect to x, we get
f'(x) = cos(x), f'(0) = 1
f''(x) = -sin(x), f'(0) = 0
f'''(x) = -cos(x), f'(0) = -1
[tex]f^{iv}(x)[/tex] = cos(x), f'(0) = 0
Maclaurin series for sin(x) becomes,
[tex]f(x) = 0 +\frac{1}{1!}x +0+(-\frac{1}{3!} )x^{3} +....[/tex]
⇒ [tex]f(x)=x-\frac{x^{3} }{3!} +\frac{x^{5} }{5!}+.....[/tex]
Now, consider [tex]f(x) = (1-x)^{-1}[/tex]
Then, the derivatives of f(x) with respect to x, we get
[tex]f'(x) = (1-x)^{-2}, f'(0) = 1[/tex]
[tex]f''(x) = 2(1-x)^{-3}, f''(0) = 2[/tex]
[tex]f'''(x) = 6(1-x)^{-4}, f'''(0) = 6[/tex]
[tex]f^{iv} (x) = 24(1-x)^{-5}, f^{iv}(0) = 24[/tex]
Maclaurin series for (1-x)^-1 becomes,
[tex]f(x) = 1 +\frac{1}{1!}x +\frac{2}{2!}x^{2} +(\frac{6}{3!} )x^{3} +....[/tex]
⇒ [tex]f(x)=1+x+x^{2} +x^{3} +......[/tex]
Thus the Maclaurin series for [tex]f(x)=sin(x)(\frac{1}{1-x} )[/tex] is
⇒ [tex]f(x)=(x-\frac{x^{3} }{3!} +\frac{x^{5} }{5!}+..... )(1+x+x^{2} +x^{3} +......)[/tex]
⇒ [tex]f(x)=x+x^{2} +x^{3} - \frac{x^{3} }{6} +x^{4}-\frac{x^{4} }{6} +.....[/tex]
⇒ [tex]f(x)=x+x^{2} +\frac{5x^{3} }{6} +\frac{5x^{4} }{6} +.....[/tex]
Hence we can conclude that the terms through degree four of the Maclaurin series is [tex]f(x)=x+x^{2} +\frac{5x^{3} }{6} +\frac{5x^{4} }{6} +.....[/tex].
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if you rolled two dice what is the probability that you would roll a 4 or a 5
HELP ASAP!!! 40 POINTS
Identify each x-value at which the slope of the tangent line to the function f(x) = 0.2x2 + 5x − 12 belongs to the interval (-1, 1).
Using derivatives, it is found that the x-values in which the slope belong to the interval (-1,1) are in the following interval:
(-15,-10).
What is the slope of the tangent line to a function f(x) at point x = x0?It is given by the derivative at x = x0, that is:
[tex]m = f^{\prime}(x_0)[/tex].
In this problem, the function is:
[tex]f(x) = 0.2x^2 + 5x - 12[/tex]
Hence the derivative is:
[tex]f^{\prime}(x) = 0.4x + 5[/tex]
For a slope of -1, we have that:
0.4x + 5 = -1
0.4x = -6
x = -15.
For a slope of 1, we have that:
0.4x + 5 = 1.
0.4x = -4
x = -10
Hence the interval is:
(-15,-10).
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What is the value of the expression when n = 3?
StartFraction 6 (n squared plus 2) Over n EndFraction
16
22
30
66
Answer:
B) 22
Step-by-step explanation:
The expression is:
[tex]\cfrac{6(n^2+2)}{n}[/tex]Find its value when n = 3, substitute n with 3 in the expression:
[tex]\cfrac{6(3^2+2)}{3} =\cfrac{6(9+2)}{3} =\cfrac{6(11)}{3}=2(11) = 22[/tex]The matching answer choice is B.
One positive integer is 2 times another positive integer and their product is 50. what are the positive integers?
The first integer is 5.
The second integer is 10.
What is Positive integer ?If an integer is higher than zero, it is positive; if it is lower than zero, it is negative. Zero can be either positive or negative. Since a b and c d, then a + c b + d, the ordering of integers is consistent with algebraic operations.
According to the information:One integer is twice the other
so,
If one integer is x then
The other will be 2x.
Their product :
2x * x = 50
2x² = 50
x² = 50/2
x² = 25
x = √25
x = 5
the first integer is 5.
the second integer is 2x = 2 x 5
= 10.
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If p(x,3) Q(7,1) and pQ(15) unit find the possible value of x
Answer:
Step-by-step explanation: P(x 3), Q(7, -1) and PQ= 5 .
To Find :
The possible value of x.
Solution :
We know, distance between two points in coordinate plane is given by :
Therefore, the possible value of x are 10 and 4.
Build Your Math Skills 2A, Round decimals to the nearest hundredth (0.01): 42.988
Answer: 42.99
Step-by-step explanation:
42.988. Since 8 is close to ten and is in the hundredth we round it hence 42.99.
Li Juan solves the equation below by first squaring both sides of the equation. [tex]\sqrt{3-2w}=w+6[/tex]
What extraneous solution does Li Juan obtain?
Answer:
w = -11
Step-by-step explanation:
[tex]\sqrt{3 - 2w} = w + 6[/tex]
[tex](\sqrt{3 - 2w})^2 = (w + 6)^2[/tex]
[tex] 3 - 2w = w^2 + 12w + 36 [/tex]
[tex] w^2 + 14w + 33 = 0 [/tex]
[tex] (w + 11)(w + 3) = 0 [/tex]
[tex] w + 11 = 0 [/tex] or [tex] w + 3 = 0 [/tex]
[tex] w = -11 [/tex] or [tex] w = -3 [/tex]
When you square both sides of an equation, you must check all solutions for extraneous solutions.
Check w = -11.
[tex]\sqrt{3 - 2w} = w + 6[/tex]
[tex] \sqrt{3 - 2(-11)} = -11 + 6 [/tex]
[tex] \sqrt{3 + 22} = -5 [/tex]
[tex] \sqrt{25} = -5 [/tex]
[tex] 5 = -5 [/tex]
This is a false statement, so the solution w = -11 is extraneous since it does not satisfy the original equation.
Check w = -3.
[tex]\sqrt{3 - 2w} = w + 6[/tex]
[tex] \sqrt{3 - 2(-3)} = -3 + 6 [/tex]
[tex] \sqrt{3 + 6} = 3 [/tex]
[tex] \sqrt{9} = 3 [/tex]
[tex] 3 = 3 [/tex]
This is a true statement, so the solution w = -3 is valid.
Answer: w = -11
Given that log 2 = 0.3010 and log 3 = 0.4771 , how can we find log 6 ?
Step-by-step explanation:
log 6 = log (2×3) = log 2 + log 3 = 0.3010+0.4771
=0.7781
Answer:
[tex]\sf \log_{10}6=0.7781[/tex]
Step-by-step explanation:
Given:
[tex]\sf \log_{10} 2 = 0.3010[/tex]
[tex]\sf \log_{10} 3 = 0.4771[/tex]
To find log₁₀ 6, first rewrite 6 as 3 · 2:
[tex]\sf \implies \log_{10}6=\log_{10}(3 \cdot 2)[/tex]
[tex]\textsf{Apply the log product law}: \quad \log_axy=\log_ax + \log_ay[/tex]
[tex]\implies \sf \log_{10}(3 \cdot 2)=\log_{10}3+\log_{10}2[/tex]
Substituting the given values for log₁₀ 3 and log₁₀ 2:
[tex]\begin{aligned} \sf \implies \log_{10}3+\log_{10}2 & = \sf 0.4771+0.3010\\ & = \sf 0.7781 \end{aligned}[/tex]
Therefore:
[tex]\sf \log_{10}6=0.7781[/tex]
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What must be true in order to find a sum for an infinite geometric series?
Answer:
0 < r < 1
Step-by-step explanation:
That every sequence gets multiplied by a number less than 1 but more than 0
The law is
Common ratio or r must lies in between 0 and 1
It means
r is in (0,1)
r is greater than 0 but less than 1The formula is
[tex]\boxed{\sf S_{\infty}=\dfrac{a}{1-r}}[/tex]
Russell has $38 and is saving $2 per day. cornelius has $64 and is spending $2 per day. after how many days will russell have more money than cornelius?
The equation be 38 + 2x - (64 - 2x) = 0 then the value of x = 6.5.
How to find the value of x?To estimate the value of x, bring the variable to the left side and bring all the remaining values to the right side. Simplify the values to estimate the result.
Let the day be x
38 + 2x - (64 - 2x) = 0
Subtract 38 from both sides
38 + 2x - (64 - 2x) - 38 = 0 - 38
Simplifying the above equation, we get
2x - (64 - 2 x) = -38
Expanding the above equation, 2x - (64 - 2x) = 4x - 64
4x - 64 = -38
Add 64 to both sides
4x - 64 + 64 = -38 + 64
Simplify
4x = 26
Divide both sides by 4
[tex]$\frac{4 x}{4}=\frac{26}{4}[/tex]
Simplifying the equation
[tex]$x=\frac{13}{2}[/tex]
The value of x = 6.5.
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Make a list of 4 different numbers whose average works up to 50
Lori was paid a 15 percent commission for the home audio system that she sold. The electronics store had marked up the price of the system by 60 percent over the wholesale price of $264. How much did Lori earn for the sale of the home audio system? $15.84 $23.76 $39.60 $63.36
Answer:
$63.36
Step-by-step explanation:
The home audio system costs $264 wholesale.
The system's price had been increased by 60 percent at the electronics retailer.
Consequently, the increased cost is 60/100x264=$158.40.
The audio system's total cost is now $422.40 (264 + 158.40).
Given that Lori received a commission of = 15% on $422.40, her commission is now (15/100) x 422.40
= $63.36
Which of the following is a monomial? O A. 11x-9 OB. 20x9 OC. 20x⁹ - 7x O D.9/x
Answer:
B. 20x⁹
Step-by-step explanation:
A monomial is a polynomial with just one term.
Example: 3x²
A. 11x - 9
INCORRECT, this has two terms 11x and 9
B. 20x⁹
CORRECT, this only has one term!
C. 20x⁹ - 7x
INCORRECT, this has two terms 20x⁹ and 7x
D.9/x
INCORRECT, this is not a polynomial
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