(a.i) If [tex]a_1,a_2,a_3,\ldots[/tex] are in arithmetic progression, then there is a constant [tex]d[/tex] such that
[tex]a_{n+1} - a_n = d[/tex]
for all [tex]n\ge1[/tex]. In other words, the difference [tex]d[/tex] between any two consecutive terms in the sequence is always the same.
[tex]a_2-a_1 = a_3-a_2 = a_4-a_3 = \cdots = d[/tex]
Now, we can expand the target expression into partial fractions.
[tex]\dfrac1{a_na_{n+1}} = \dfrac{\alpha}{a_n} + \dfrac{\beta}{a_{n+1}}[/tex]
Combining the fractions on the right and using the recursive equation above, we have
[tex]\dfrac1{a_na_{n+1}} = \dfrac{\alpha (a_n + d) + \beta a_n}{a_n(a_n+d)} = \dfrac{(\alpha+\beta) a_n + \alpha d}{a_n a_{n+d}} \\\\ \implies \begin{cases}\alpha + \beta = 0 \\ \alpha d = 1 \end{cases} \implies \alpha = \dfrac1d, \beta = -\dfrac1d[/tex]
and hence
[tex]\dfrac1{a_n a_{n+1}} = \dfrac1{da_n} - \dfrac1{da_{n+1}}[/tex]
as required.
(a.ii) Using the previous result, the [tex]n[/tex]-th term [tex](n\ge1)[/tex] in the sum on the left is
[tex]\dfrac1{a_n a_{n+1}} = \dfrac1d \left(\dfrac1{a_n} - \dfrac1{a_{n+1}}\right)[/tex]
Expand each term in this way to reveal a telescoping sum:
[tex]\dfrac1{a_1a_2} + \dfrac1{a_2a_3} + \dfrac1{a_3a_4} + \cdots + \dfrac1{a_{99}a_{100}} \\\\ ~~~~~~~~ = \dfrac1d \left(\left(\dfrac1{a_1} - \dfrac1{a_2}\right) + \left(\dfrac1{a_2} - \dfrac1{a_3}\right) + \left(\dfrac1{a_3} - \dfrac1{a_4}\right) + \cdots + \left(\dfrac1{a_{99}} - \dfrac1{a_{100}}\right)\right) \\\\ ~~~~~~~~ = \dfrac1d \left(\dfrac1{a_1} - \dfrac1{a_{100}}\right) = \dfrac{a_{100} - a_1}{d a_1 a_{100}}[/tex]
By substitution, we can show
[tex]a_n = a_{n-1} + d = a_{n-2} + 2d = \cdots = a_1 + (n-1)d \\\\ \implies a_{100} = a_1 + 99d[/tex]
so that the last expression reduces to
[tex]\dfrac{(a_1 + 99d) - a_1}{d a_1 a_{100}} = \dfrac{99d}{d a_1 a_{100}} = \dfrac{99}{a_1 a_{100}}[/tex]
as required. More generally, it's easy to see that
[tex]\dfrac1{a_1a_2} + \dfrac1{a_2a_3} + \dfrac1{a_3a_4} + \cdots + \dfrac1{a_na_{n+1}} = \dfrac{n}{a_1a_{n+1}}[/tex]
(b) I assume you mean the equation
[tex]\dfrac1{3\times7} + \dfrac1{7\times11} + \dfrac1{11\times15} + \cdots + \dfrac1{k(k+4)} = \dfrac2{25}[/tex]
Note that the distinct factors of each denominator on the left form an arithmetic sequence.
[tex]a_1 = 3[/tex]
[tex]a_2 = 3 + 4 = 7[/tex]
[tex]a_3 = 7 + 4 = 11[/tex]
and so on, with [tex]n[/tex]-th term
[tex]a_n = 3 + (n-1)\times4 = 4n - 1[/tex]
Let [tex]a_n=k[/tex]. Using the previous general result, the left side reduces to
[tex]\dfrac1{3\times7} + \dfrac1{7\times11} + \dfrac1{11\times15} + \cdots + \dfrac1{a_na_{n+1}} = \dfrac n{3a_{n+1}} \\\\ \implies \dfrac{\frac{k+1}4}{3(k+4)} = \dfrac2{25}[/tex]
Solve for [tex]k[/tex].
[tex]\dfrac{k+1}{12k+48} = \dfrac2{25} \implies 25(k+1) = 2(12k+48) \\\\ \implies 25k + 25 = 24k + 96 \implies \boxed{k=71}[/tex]
h(t)=t^4+3t^2-t+3
h'?
Answer: [tex]h'(t)=4t^3 + 6t-1[/tex]
Step-by-step explanation:
Use the power rule for differentiation.
Find the total surface area of the cylinder shown. Leave the answer in terms of pi radius 4 height 6
Answer: [tex]80\pi[/tex]
Step-by-step explanation:
- the equation for the surface area of a cylinder is [tex]A=2\pi rh+2\pi r^{2}[/tex]
- the equation for this cylinder will be [tex]A=2\pi (4)(6)+2\pi (4^{2})\\[/tex]
- so, [tex]2\pi (4)(6) = 150.7964474[/tex] and [tex]2\pi (4^{2})\\ = 150.7964474[/tex]
- then [tex]150.7964474 + 150.7964474= 251.3274123[/tex]
- finally, [tex]251.3274123/\pi = 80[/tex]
- so the answer in terms of [tex]\pi[/tex] is [tex]80\pi[/tex]
hope this helps :)
Help plsss there’s a timer
Answer:
[tex]a=27, b=81[/tex]
Step-by-step explanation:
Plug in the values
[tex]a=3^3\\b=3^4\\\\a=27\\b=81[/tex]
The equation y minus 1 = negative 7 (x minus 3). is written in point-slope form. What is the y-intercept of the line?
Answer:
(0, 22)
Step-by-step explanation:
So you have the equation:
[tex]y-1=-7(x-3)[/tex]
So first let's define what the "y-intercept" is. The y-intercept, of a line is like the name implies, when the line intercepts or "passes" the y-axis. And if you were to take any point on the y-axis, the y-value may vary, but there is one value that doesn't change, and that's the x-value. the x-value is always going to be zero, no matter where you are on the y-axis. So to find the y-intercept simple plug 0 in as x. This gives you the equation:
[tex]y-1 = -7(0-3)[/tex]
[tex]y-1 = -7(-3)[/tex]
[tex]y-1=21[/tex]
[tex]y=22[/tex]
So the y-intercept is at (0, 22)
4(4 - x) - 8(-2 + 4x)
If you want the answer simplified, its
-36x + 32
Consider the rectangle enlargement. a smaller rectangle has a length of 4.3 millimeters and width of 4.5 millimeters. a larger rectangle has a length of x millimeters and width of 24.3 millimeters. the perimeter of the original rectangle is . the side length of the enlarged rectangle is . the perimeter of the enlarged rectangle is .
The perimeter of the original rectangle is 17.6mm. The side length of the enlarged rectangle is 23.22mm and the perimeter of the enlarged rectangle is 95.04mm
Perimeter of a rectangleThe formula for calculating the perimeter of a rectangle is expressed as;
P = 2(l +w)
If the smaller rectangle has a length of 4.3 millimeters and width of 4.5 millimeters while a larger rectangle has a length of x millimeters and width of 24.3 millimeters, then;
4.3/x = 4.5/24.3
Cross multiply
4.5x = 104.49
x = 23.22mm
Perimeter of original rectangle = 2(4.3 + 4.5)
Perimeter of original rectangle = 17.6mm
Perimeter of enlarged rectangle = 2(23.22 + 24.3)
Perimeter of enlarged rectangle = 95.04mm
The perimeter of the original rectangle is 17.6mm. The side length of the enlarged rectangle is 23.22mm and the perimeter of the enlarged rectangle is 95.04mm
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Answer:
17.6mm, 23.22mm, 95.04mm
Step-by-step explanation:
Graph the line with slope -3 passing through the point(-3,-2) .
The attached graph represents the graph of the line
How to plot the graph?The given parameters are:
Slope, m = -3
(x1, y1) = (-3, -2)
The equation is calculated as:
y = m(x - x1) + y1
So, we have:
y = -3(x + 3) - 2
Expand
y = -3x - 9- 2
Evaluate the difference
y = -3x - 11
See attachment for the graph of y = -3x - 11
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Form the table of backward differences for the function f(x) = x³ +5x-7 for x = -1, 0, 1, 2, 3, 4 5 and 6.
The table of values is
x -1 0 1 2 3 4 5 6
f(x) -13 -7 -1 11 35 77 143 139
How to form the table?The function is given as:
f(x) = x³ +5x-7
And the values of x are
x = -1, 0, 1, 2, 3, 4 5 and 6.
Substitute these values in the function
f(-1) = (-1)³ +5(-1) - 7 = -13
f(0) = (0)³ +5(0) - 7 = -7
f(1) = (1)³ +5(1) - 7 = -1
f(2) = (2)³ +5(2) - 7 = 11
f(3) = (3)³ +5(3) - 7 = 35
f(4) = (4)³ +5(4) - 7 = 77
f(5) = (5)³ +5(5) - 7 = 143
f(6) = (6)³ +5(6) - 7 = 239
Hence, the table of values is
x -1 0 1 2 3 4 5 6
f(x) -13 -7 -1 11 35 77 143 139
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Write -3.65 as a mixed number.
Answer:
-3 13/20
Step-by-step explanation:
-3.65=
-3 65/100=
-3 13/20
Giving brainliest :)
Answer:
I think it would be D the last answer on the page I am pretty confident in the answer but if for some reason it's wrong then I'm sorry.
Step-by-step explanation:
The sum of 8 times a number and 10 is 34. Find the number.
Answer: 3
8x+10=34
sub 10
8x=24 divide by 8
x=3
Step-by-step explanation:
2+2*+456+5367+63784+464749=
Answer:534360
Step-by-step explanation:
I did have to remove the multiplication though
8. determine the shape:
Melissa: Does your shape have exactly one right angle?
Keya: Yes
Melissa: Does your shape have opposite congruent sides?
Keya: No
Melissa: Does your shape have any parallel sides?
Keya: Yes
Melissa: The shape is a _____________________
How did you rule out other shapes?
b) Michelle: Does your shape have 2 pairs of congruent opposite sides?
Olga: Yes
Michelle: Does your shape have right angles?
Olga: No
Michelle: Does your shape have 2 pairs of opposite parallel sides?
Olga: Yes
What is the shape?
Is there an additional question Michelle should ask to be sure? If so, what should she ask?
Quadrilaterals are plane shapes with four sides. Thus, the required answers are:
a) The first shape is a right trapezoid.
b) The second shape is a parallelogram.
The given properties are that of a quadrilateral, thus both shape is a quadrilateral. A quadrilateral is a family of shapes that have four straight sides. Examples include squares, rhombus, kites, etc.
A. Given that the shape has one right angle and parallel sides. Then considering the given properties, the most likely shape is a right trapezoid.
A right trapezoid is a type of trapezium that has one right angle, and a pair of parallel sides. Thus the shape here is a right trapezoid.
Other shapes could be ruled out by comparing their properties with the properties of the shape given in the question.
B. The properties of the required shape as given are: it has two pairs of congruent opposite sides, and 2 pairs of opposite parallel sides. Thus the most likely shape required here is a parallelogram. A parallelogram is a shape with parallel and congruent opposite sides, and no right angle.
Other shapes could be ruled out by comparing their properties with the properties of the shape given in the question.
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Kindly contact a 1-on-1 tutor if more explanations are required.
Gathering information from individuals in a sample is known as _______________. A. data dumping B. data collection C. data diving D. data displaying
Gathering information from individuals in a sample is known as : Data collection. ( B )
What is statistics?Statistics is the science of collecting and analyzing data in quantities, especially for the purpose of informing.
Analysis:
sample collection is the first approach to analyzing any information.
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la mitad mas el tercio mas la cuarta parte de los años que tiene manuel, suman los años que tiene mas tres ¿que edad tiene manuel? ·
Resolviendo una ecuación lineal veremos que que Manuel tiene 36 años.
¿Qué edad tiene Manuel?Vamos a definir la edad de Manuel como x.
Sabemos que la mitad mas el tercio mas la cuarta parte de x es igual que x más 3, entonces:
x/2 + x/3 + x/4 = x + 3
Ahora podemos resolver esta ecuación lineal para x:
x/2 + x/3 + x/4 - x = 3
x*( 1/2 + 1/3 + 1/4 - 1) = 3
x = 3/(1/2 + 1/3 + 1/4 - 1) = 36
Así, concluimos que manuel tiene 36 años.
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Find a formula for the general term an of the sequence, assuming that the pattern of the first few terms continues. (Assume that n begins with 1.)
{2, 6, 10, 14, 18, . . . }
The formula for the general term of the sequence is Tn = 4n - 2
How to determine the formula?The sequence is given as:
{2, 6, 10, 14, 18, . . . }
The above sequence is an arithmetic sequence with the following features:
First term, a = 2Common difference, d = 4 i.e. 6 - 2The formula for the general term is calculated using:
Tn = a + (n - 1) * d
This gives
Tn= 2 + (n - 1) * 4
Expand
Tn = 2 + 4n - 4
Evaluate the like terms
Tn = 4n - 2
Hence, the formula for the general term of the sequence is Tn = 4n - 2
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There are chocolates in a box, all identically shaped. There are 8 filled with nuts, 16 with caramel, and 14 are solid chocolate. You randomly select one piece, eat it, and then select a second piece. Find the probability of selecting 2 solid chocolates in a row.
Answer:
P = [tex]\frac{91}{703}[/tex] ≅ 12.95%.
Step-by-step explanation:
The total amount of chocolates in the box is 8 + 16 + 14 = 38, 14 of which are solid.
Let's calculate the probability of getting a solid chocolate in the first selection:
[tex]P(1S) = \frac{14}{38} = \frac{7}{19}.[/tex]
Given that the first piece was solid, there are now 37 pieces of chocolate in the box, 13 of which are solid. Let's now calculate the probability of selecting another solid chocolate:
[tex]P(2S\,|\,1S) = \frac{13}{37}.[/tex]
Thus, the probability of selecting two solid chocolates in a row is the product of the two probabilities calculated above:
[tex]P = \frac{7}{19}\,.\,\frac{13}{37} = \frac{91}{703}.[/tex]
I have tried this problem multiple times now and don't get the correct answer. Please help!
The confidence interval is (-1.1059, 7.70459).
What is confidence interval in statistics?Image result for what is confidence interval
A confidence interval is the mean of your estimate plus and minus the variation in that estimate.
The sample mean is M=3.3
The sample size is N=44
So, σM= s/√N = 16.6/ √44
= 16.6/ 6.633
= 2.621
and, the degree of freedom
df= n-1
=44-1
=43
So, The t-value for a 90% confidence interval and 43 degrees of freedom is t=1.681.
Now, Margin of error = 2.261* 1.681 = 4.4059
Now, the lower and upper bound of the confidence interval are:
LL = 3.3- 4.40459 = -1.1059
UL = 3.3+ 4.40459 = 7.70459
Hence, the confidence interval is (-1.1059, 7.70459).
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The lateral surface area of cone A is equal to the lateral surface area of cylinder B.
True or False?
Answer:
True.
Step-by-step explanation:
Lateral Surface area of cone=πrl=πr×2h=2πrh
Lateral surface area of cylinder=2πrh
FAST HELPPPPPPP
The point at which three or more lines intersect is the point of
A.equality.
B.concurrency.
C.parallelism.
D.tangency.
Answer:
B.concurrency.
Step-by-step explanation:
Three or more lines when met at a single point are said to be concurrent and the point of intersection is point of concurrency.
465 draws will be made at random with replacement from the box [0 0 1]. The number of 1’s among the draws will be around…
The number of 1’s among the draws will be 155.
How to find that a given condition can be modelled by binomial distribution?Binomial distributions consist of n independent Bernoulli trials.
The expected value is given as
E(X) = np
465 draws will be made at random with replacement from the box [0 0 1].
The number of 1’s among the draws will be
p = 1/3
n = 465
Then the expected value is given as
E(x) = 465 × 1/3
E(x) = 155
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6(m + 2 + 7m)
Part A: Write two expressions that are equivalent to the given expression. (3 points)
Part B: Show that one of your expressions in Part A is equivalent to the given expression using algebraic properties. Explain which properties you used. (4 points)
Part C: Show that your other expression from Part A is equivalent to the given expression by substituting a number for m. (3 points)
Part A: 6 ( 8m + 2) and 48m + 12
Part B: 48m - 48m = 12 - 12
Part C: 6 (10) = 6(10)
How to determine the expressions1. The two expressions could be gotten thus;
6(m + 2 + 7m)
Collect like terms in the bracket
6 ( 8m + 2) ⇒ first expression
6(m + 2 + 7m)
Expand the bracket
6m + 12 + 42m
Collect like terms
6m + 42m + 12
48m + 12 ⇒ second expression
Part B:
6(m + 2 + 7m) = 48m + 2
Expand the bracket
6m + 12 + 42m = 48m + 2
Collect like terms
48m - 48m = 12 - 12
0 = 0
Part C:
Let the expression from A be 6 ( 8m + 2)
6(m + 2 + 7m) = 6 ( 8m + 2)
Let the number be m = 1
Substitute the values
6(1 + 2 + 7(1) ) = 6 (8(1) + 2)
6 ( 1 + 2 + 7) = 6 (10)
6 (10) = 6(10)
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If 7 spiders make 7
webs in 7 days, then 1
spider will make 1
web in how many
days?
Answer:
1 day
Step-by-step explanation:
since 7 spider =7 days
1 spider =? day
cross multiply
7 days × 1 spider = 7 spider × ? days
7 spider = 7 spider × ? days
7 spider 7 spider
= 1 days
Latesha justifies whether the expression 4x – 3 is equivalent to one-half (4 x minus 6) minus 2 x (1 minus 3) by letting x = 2 in both expressions. which values should latesha compare to justify whether the expressions are equivalent? 2 and 5 5 and 9 5 and 5 9 and 9
Answer:
5 and 9
Step-by-step explanation:
Answer:b
Step-by-step explanation: took the test
the sum of two numbers is 24 and the quotient is 3. what are the to numbers
Answer:
18 and 6
Step-by-step explanation:
18+6=24
18/6=3
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Can someone answer this
We can see that the unit rate of y = 6.5*x is larger than the unit rate on the graph.
How to get the unit rate of the line?
The unit rate is the equal to the slope of the line.
In the written line, y = 6.5*x, the unit rate is 6.5
On the graphed line we can see that it passes through the points (0, 0) and (1, 3), so the unit rate is:
[tex]r = \frac{3 - 0}{1 - 0} = 3[/tex]
Then we can see that the unit rate of y = 6.5*x is larger than the unit rate on the graph.
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Complete the identity: sec(t) - cos(t) = ?
1) sec(t)csc(t)
2) -2tan²(t)
3) 1 + cot(t)
4) sin(t)tan(t)
Step-by-step explanation:
[tex] \sec( \alpha ) - \cos( \alpha ) [/tex]
[tex] \frac{1}{ \cos( \alpha ) } - \cos( \alpha ) [/tex]
[tex] \frac{1}{ \cos( \alpha ) } - \frac{ \cos {}^{2} ( \alpha ) }{ \cos( \alpha ) } [/tex]
[tex] \frac{ \sin {}^{2} ( \alpha ) }{ \cos( \alpha ) } [/tex]
[tex] \frac{ \sin( \alpha ) }{ \cos( \alpha ) } \times \frac{ \sin( \alpha ) }{1} [/tex]
[tex] \tan( \alpha ) \sin( \alpha ) [/tex]
4 is the option
Which type of parent function is f(x) = -
1/?
O A. Absolute value
OB. Reciprocal
OC. Square root
OD. Linear
Answer: B. Reciprocal
Step-by-step explanation:
This is a fact.
The parent function represented by f(x) = -1/x is the reciprocal function.
Therefore, the correct answer is option B.
Reciprocal functions, also known as rational functions, are defined by an algebraic expression where the dependent variable is in the denominator. In this case, the dependent variable is x, and it appears in the denominator as 1/x.
The negative sign in front of the fraction simply indicates that the function is reflected over the x-axis.
The general form of a reciprocal function is f(x) = a/x, where "a" is a constant. In the given function f(x) = -1/x, a is equal to -1.
In the case of f(x) = -1/x, the graph of the function is a hyperbola that opens downwards in the first and third quadrants. It approaches the x-axis as x moves towards positive or negative infinity, and it moves away from the x-axis as x approaches 0.
Therefore, the correct answer is option B, the reciprocal function.
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If y is inversely proportional to x and y=−54 when x=38, find y if x=30. (Round off your answer to the nearest hundredth.)
Answer:
-68.4
Step-by-step explanation:
answer explained on the photo !
What are the vertex and x-intercepts of the graph of y=x² - 2x - 24? Select
one answer for the vertex and one for the x-intercepts.
A. x-intercepts: (4,0), (-6, 0)
B. x-intercepts: (-4,0), (-6, 0)
C. Vertex: (1, -25)
D. Vertex: (1, 23)
E. Vertex: (-1, -21)
OF. x-intercepts: (-4, 0), (6, 0)
Answer:
C. Vertex (1;-25)
F. x-intercepts: (-4;0), (6;0).
Step-by-step explanation:
1) x-coordinate of the vertex: -b/2a=1; y-coordinate of the vertex: -25;
2) for x-intercepts it needed to solve the equation x²-2x-24=0;
x=6;-4;
3) finally, correct answers are: C. Vertex (1;-25) and F. x-intercepts: (-4;0), (6;0).