A6-Sequences and Series
Q1. Below are the first few terms of some important mathematical sequences. State the 7th term
of each sequence:
a) 2, 4, 6, 8, .......
b) 1, 4, 9, 16, ....
c) 1, 8, 27, 64, ....
d) 2, 3, 5, 7, 11, ....


7th term is.
7th term
7th term is .............. (1)
7th term is ........…….…….. (1)
(Total 4 marks)

A common inhabitant of human intestines is the bacterium Escherichia coli, named after the German pediatrician the odor Escherich, who identified it in 1885. A cell of this bacterium in a nutrient-broth medium divides into two cells every 20 minutes. The initial population of a culture is 50 cells.

The relative growth rate is k = 3 ln 2 hr -1

To find out the relative growth rate:

Let the exponential model is

A = [tex]Pe^{kt}[/tex]

We have been given that

(a)Cell divides into two cells every 20 minutes

[tex]t = \frac{20}{60} = \frac{1}{3}hr[/tex]

A = 2P

Thus, we have

2P= Pe[tex]\frac{k}{3}[/tex]

In 2 = In(e[tex]\frac{k}{3}[/tex])

In 2 = [tex]\frac{k}{3}[/tex]

k = 3 ln 2 hr -1

Hence, the answer is k = 3 ln 2 hr -1 for the relative growth rate

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The answer is 12x-1

From the question, we have

2(3x - 5) +3 (2x+3)

=6x-10+6x+9

=12x-1

The answer is 12x-1

Addition:

The act of combining two or more items is known as addition. The process of computing the sum of two or more numbers in mathematics is called addition. It is a fundamental mathematical procedure that is frequently used in daily life. When we work with money, figure out our food bills, or figure out the time, we frequently employ addition. One-digit numbers can be added simply for solving addition problems, however for bigger numbers, we divide the numbers into columns using their corresponding place values, such as ones, tens, hundreds, thousands, and so on.

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The positive value of sin x/2 is [tex]\frac{\sqrt{30} }{10}[/tex]

Trigonometric ratios are the ratios of sides of a right-angle triangle. The most common trigonometric ratios are sine, cosine, and tangent.

The relationship between cos x and sin x/2 is

sin x/2 = [tex]\sqrt{\frac{1 - cosx}{2} }[/tex]

cos x = 2/5

sin x/2 = [tex]\sqrt{\frac{1-\frac{2}{5} }{2} }[/tex]

1 - 2/5 = 3/5

simplifying 3/5 ÷ 2

removing division sign and taking the reciprocal of 2, we have

3/5 x 1/2 = 3/10

so that

sin x/2 = [tex]\sqrt{\frac{3}{10} }[/tex]

which can be written as

sin x/2 = [tex]\frac{\sqrt{3} }{\sqrt{10} }[/tex]

By rationalizing   [tex]\frac{\sqrt{3} }{\sqrt{10} }[/tex] we multiply it with [tex]\frac{\sqrt{10} }{\sqrt{10} }[/tex]

so that we have  [tex]\frac{\sqrt{3} }{\sqrt{10} }[/tex] x  [tex]\frac{\sqrt{10} }{\sqrt{10} }[/tex] = [tex]\frac{\sqrt{30} }{10}[/tex]

In conclusion, the radical form of sin x/2 is  [tex]\frac{\sqrt{30} }{10}[/tex]

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The point estimate of the proportion of vehicles not covered by insurance is 0.23.

Given x = Number of Successes = 69

n = sample size = 300

We are interested in the point estimate of the population proportion p.

What is the point estimate of the population proportion p and how do we calculated is shown below.

The point estimate of the number of successes x and the sample size n

p = x/n

Substituting the known values into the formula and evaluating, we obtain:

p = x/n = 69/300 = 0.23

Hence the answer is The point estimate of the proportion of vehicles not covered by insurance is 0.23.

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Answer:

(6,3)

Step-by-step explanation:

A big sample that should the experimenter take from the population is 256 and if the standard deviation and the width of the confidence interval are both doubled then the sample is also 256.

In the given question,

The confidence level = 99%

Given width = 0.5

Standard deviation of resistance(σ)= 1.55

The formula to find the a big sample that should the experimenter take from the population is

Margin of error(ME) =[tex]z_{\alpha /2}\frac{\sigma}{\sqrt n}[/tex]

So n= [tex](z_{\alpha /2}\frac{\sigma}{ME})^2[/tex]

where n=sample size

We firstly find the value of ME and [tex]z_{\alpha /2}[/tex].

Firstly finding the value of ME.

ME=Width/2

ME=0.5/2

ME=0.25

Now finding the value of [tex]z_{\alpha /2}[/tex].

The given interval is 99%=99/100=0.99

The value of [tex]\alpha[/tex] =1−0.99

The value of [tex]\alpha[/tex] =0.01

Then the value of [tex]\alpha /2[/tex] = 0.01/2 = 0.005

From the standard table of z

[tex]z_{0.005}[/tex] =2.58

Now putting in the value in formula of sample size.

n = [tex](2.58\times\frac{1.55}{0.25})^2[/tex]

Simplifying

n=(3.999/0.25)^2

n=(15.996)^2

n=255.87

n≈256

Hence, the sample that the experimenter take from the population is 256.

Now we have to find the sample size if the standard deviation and the width of the confidence interval are both doubled.

The new values,

Standard deviation of resistance(σ)= 2×1.55

Standard deviation of resistance(σ)= 3.1

width = 2×0.5

width = 1

Now the value of ME.

ME=1/2

ME=0.5

The z value is remain same.

Now putting in the value in formula of sample size.

n= [tex](2.58\times\frac{3.1}{0.5})^2[/tex]

Simplifying

n=(7.998/0.5)^2

n=(15.996)^2

n=255.87

n≈256

Hence, if the standard deviation and the width of the confidence interval are both doubled then the sample size is 256.

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Answer:

5 and 9

Step-by-step explanation:

3(x+y) = 42

5(x-y) = 20

3/3 (x+y) = 42/3

5/5 (x-y) = 20/5

x+y = 14

x-y = 4

x-x = 0

y-(-y) = y+y = 2y

14-4 = 10

2y = 10

2/2 y = 10/2

y = 5

x-5 = 4

x = 4 + 5

x = 9

Answer:

One number is 5 and the other number is 9

Step-by-step explanation:

Answer: its (4f 5) ^3

Step-by-step explanation:

Answer:

It should be 80.

Answer:

The 5th row has 24 chairs, and therefore not enough chairs for 100 people.

Step-by-step explanation:

The equation for this situation is , f(x) =  4x+4  The reason for this is, there is 4 chairs added to every row, so in order to make row number one make sense, you must subtract 4 to find the y-intercept which is also 4.

To find if the 5th row has enough for 100 people, you must plug 5 in for x

The 5th row has 24 chairs, and therefore not enough chairs for 100 people.

The required chairs in the 5th row there is 24 chairs and 5 rows are not enough to accommodate 100 people.

Arithmetic progression is the series of numbers that have common differences between adjacent values are common.

Here,
A given scenario is of arithmetic progression,
Let the first term be a = 8,
and the common difference between chairs is d = 4,
Now,
Let the number of rows be n,

Now,
The explicit arithmetic formula is given as,
Number of chairs = 8 + (n - 1)4
Now put n = 5
Number of chairs = 8 + (5 - 1) 4
                           = 8 + 16
                           = 24

So, in the 5th row, there will be 24 chairs.

Now,
For 100 people number of chairs would be 100
Apply the sum formula of arithmetic progression,

Where sum gives the total number of chairs for n rows,

Sum = n / 2 (a + l)
Sum = 5 / 2(8 + 24)
       = 5 / 2 (32)
       = 5 × 16
       = 80 chairs

Thus, the required chairs in the 5th row there is 24 chairs and 5 rows are not enough to accommodate 100 people.

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The distance it will reach to  x, = 46.6 feet

acc to our question-

hence,The distance it will reach to  x, = 46.6 feet

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Answer:

q=10 n=10

Step-by-step explanation:

nickel = 5¢ (or 0.05)

quarters = 25¢ (or .25)

[tex]q+n=20\\.25q+.05n=3\\\\\\-.05(q+n)=20(-.05)\\-.05q-.05n=-1\\\\-.05q-.05n=-1\\.25q+.05n=3\\---------\\0.2q= 2\\-- --\\0.2 \\q=10 \\\\10+n=20\\n=20-10\\n=10\\\\.25(10)+.05(10)=3\\2.5+0.5=3[/tex]

Michael depicts a quadrilateral with four congruent sides but no congruent angles; the generated quadrilateral is best described as a rectangle.

A rectangle is a quadrilateral with four right angles in the Euclidean plane. It can alternatively be described as a parallelogram with a right angle or an equiangular quadrilateral, where equiangular denotes that all of its angles are equal (360°/4 = 90°). A square is a rectangle with four equally long sides. Oblong is occasionally used to describe a rectangle that isn't square. Rectangle ABCD would be the designation for a rectangle with the vertices ABCD.a quadrilateral with four congruent sides but no congruent angles; the generated quadrilateral is best described as a rectangle.

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Answer:

x=2

y=16

Step-by-step explanation:

fill in y

2x+(6x+4)=20

8x+4=20

   -4.  -4

8x=16

/8. /8

x=2

Now fill in X to find y

y=6(2)+4

y=12+4

y=16

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Answer:

i know arts

Step-by-step explanation:

Answer:

$34.24

Step-by-step explanation:

Answer: 34.24

Step-by-step explanation:

the approximate percentage of daily phone  calls is 0.

Mean= 58

Standard deviation= 4

x= 54 and 62.

Now, solving to find the percentage of daily phone calls numbering between 54 and 62.

[tex]let x_{1} =54 and x_{2}=62[/tex]

first calculating the z-score for daily 54 phone calls.

Formula used:

[tex]z-score = x-mean/standard deviation[/tex]

[tex]z- score=54-58/4[/tex]

[tex]z-score=-4/4[/tex]

z-score= -1

∴ z-score for daily 54 phone call is -1.

Next, calculating the z-score for daily 62 phone calls.

z-score= 62-58/4

z-score= 4/4

z-score= 1

∴ z-score for daily 62 phone call is 1.

We can observe that there is change in z-score for 54 phone call and 62 phone call.

Lets use the normal distribution table to find the percentage of daily phone calls numbering between 54 and 62.

⇒ Percentage of daily phone calls numbering between 54and 62=

[tex](z-scorex_{1})-(z-scorex_{2} )[/tex]

⇒ Percentage of daily phone calls numbering between 54 and 62=

(z-score1) - (z-score-1)

Using normal distribution table

⇒ Percentage of daily phone calls numbering between 54 and 62=

0.34134-0.34134 = 0

Hence, 0 is the percentage of daily phone calls numbering between 54 and 62.

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We have to calculate the estimated standard error and means of the two samples before calculating  the test statistic for a two sample test of means with population variances equal but unknown.

In the given question, we have to explain steps that are necessary before you can calculate the test statistic for a two sample test of means with population variances equal but unknown.

The test statistic for a two-sample independent t-test is derived by dividing the difference between the means of the two samples by the estimated standard error, either pooled or unpooled. The total amount of variation in both groups is gauged by the anticipated standard error.

So we have to calculate the estimated standard error and means of the two samples before calculating  the test statistic for a two sample test of means with population variances equal but unknown.

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Answer:

x=6

Step-by-step explanation:

11+5x/2=26

-11.         -11

5x/2=15

x2.     X2

5x=30

/5.   /5

x=6

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The percent increase in the number of customers is  48% .

In the question ,

it is given that

number of customers came ice cream shop in first weekend  = 316 customers

number of customers that came in second weekend = 468 customers .

increase in number of customers = 468 - 316 = 152

So , the percent increase in number of customers can be calculated using the formula ,

percent increase = (increase in customer)/(number of customers in first weekend) × 100

Substituting the values , in  the percent increase formula , we get

percent increase = 152/316 × 100

= 0.4810 × 100

= 48.10

≈ 48%

Therefore , The percent increase in the number of customers is  48% .

The given question is incomplete ,the complete question is

An ice cream shop had 316 customers the first weekend it opened. The second weekend it had 468. What is the percent of increase in the number of customers? Round to the nearest percent.

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The statement about the angle of ΔUVW must be true is m∠W>m∠U>m∠V. The correct option is F.

Given that in ΔUVW, VW = 14, WU = 2, and UV = 15.

We want to find the statement about the angle ΔUVW must be true from the attached image.

As we know the theorem that, If one side of a triangle is longer than another side, then the angle opposite the longer side will be larger than the angle opposite the shorter side.

Then according to the above theorem that the long side of the triangle corresponds to the big angle theorem.

So, we get that m∠W>m∠U>m∠V

Hence, the statement which is true about the ΔUVW is m∠W>m∠U>m∠V.

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The triangle congruence theorem that could be used to show that triangle ADE is congruent to triangle CBE is Angle-Side-Angle Triangle Congruence Theorem also known as ASA Triangle Congruence Theorem

Congruence Theorem

Congruence theorem states that "if all the three sides of one triangle are equal to all the three sides of another triangle, then both the triangles are congruent to each other."

Given,

Here we need to identify the angle congruence theorem could you use to prove triangle ADE is congruent to triangle CBE.

While we looking into the given triangles we can say that;

=> ∠AED = ∠CEB

Because they are both vertically opposite angles and vertically opposite angles are equal.

From the given triangle, we are given that;

That is,

=> ∠DAE = ∠BCE

We are also know that

=> AE = CE

Therefore, this means that we have 2 corresponding angles and 1 corresponding included side that are congruent.

So, the congruence theorem that this represents is Angle-Side-Angle Congruence theorem.

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The system of equations that models the given scenario is t + u = 9 and 10t + u = 10u + t - 9.

What is an equation?

An equation is a mathematical statement that proves two mathematical expressions are equal in algebra, and this is how it is most commonly used. In the equation 3x + 5 = 14, for instance, the two expressions 3x + 5 and 14 are separated.

The answer would be "t + u = 9 and 10t + u = 10u + t - 9".

This can be found if you replaced t with 4 and u with 5.

Adding the 2 digits would give you 9.

Use the formula "10t + u = 10u + t - 9".

Replace t and u with 4 and 5 then simplify. 10*4 + 5 = 10*5 + 4 - 9. Simplifying you'd get 45 = 54 - 9 or 45 = 45 which makes the statement true.

Thus making "t + u = 9 and 10t + u = 10u + t - 9" the answer.

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Answer:

  964.6 in³

Step-by-step explanation:

You want the volume of a 12-inch cube with a 9-inch diameter hole through its center. The axis of the hole is perpendicular to opposite faces.

The volume of the hole is the volume of a cylinder 4.5 inches in radius and 12 inches high. That volume is ...

  V = πr²h

  V = π(4.5 in)²(12 in) = 243π in³ ≈ 763.4 in³

The volume of a 12-in cube is ...

  V = s³

  V = (12 in)³ = 1728 in³

The volume of the piece is the amount of the cube remaining after the volume of the hole is removed.

  piece volume = cube volume - hole volume

  piece volume = 1728 in³ -763.4 in² = 964.6 in³

The volume of the piece is about 964.6 cubic inches.

The volume of the toy  is about 406.42 cubic inches.

we want the volume of a 12-inch cube with a 9-inch diameter hole through its center. The axis of the hole is perpendicular to opposite faces.

Hole volume

The volume of the hole is the volume of a cylinder 4.5 inches in radius and [tex]\sqrt{3}[/tex] 12 inches high. ( Volume of the cylinder)

 V = πr²h

 V = π(4.5 in)²([tex]\sqrt{3}[/tex] 12 in) = 243 [tex]\sqrt{3}[/tex] π in³ ≈ 1321.58 in³

Cube volume

The volume of a 12-in cube is .

V = s³

 V = (12 in)³ = 1728 in³

piece volume = cube volume - hole volume

 piece volume = 1728 in³ - 1321.58 in² = 406.42 in³

The volume of the piece is about 406.42cubic inches.

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The better deal between the two deals on turkey leg is 5 legs for .70

The better deal is solves by comparing the 2 prices with the unit price

if 3 legs for 3.99 marks one leg will be

3 = 3.99

1 = ?

cross multiplying

3 * ? = 1 * 3.99

? = 3.99/3

? = 1.33

if 5 legs for 0.70, one leg will be

5 = 0.70

1 = ?

cross multiplying

5 * ? = 1 * 0.7

? = 0.7/5

? = 0.14

5 legs for 0.70, is cheaper per unit price and a better deal

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The correct option (A) -5 which represents the slope of the graphed line having points (0,2) and (1,-3).

The general form for forming the linear equation is:

[tex]y-y1= m (x-x1)[/tex]

where x and y are axis coordinates respectively.

m is the slope of the straight line.

Now we need to calculate the slope of the line.

Formula used:

[tex]m = y2- y1/x2-x1[/tex]

Here,

[tex](x1, y1) = (0, 2), (x2, y2) = (1,-3).[/tex]

[tex]m = (-3-2)/(1-0)\\m = -5/1\\m = -5[/tex]

Therefore, the slope of the graphed line is -5 having points (0,2) and (1,-3).

So, the -5 number best represents the slope of the graphed line.

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Step-by-step explanation:

The correct option (A) -5 which represents the slope of the graphed line having points (0,2) and (1,-3).

The general form for forming the linear equation is:

y-y1= m (x-x1)y−y1=m(x−x1)

where x and y are axis coordinates respectively.

m is the slope of the straight line.

Now we need to calculate the slope of the line.

Formula used:

m = y2- y1/x2-x1m=y2−y1/x2−x1

Here,

(x1, y1) = (0, 2), (x2, y2) = (1,-3).(x1,y1)=(0,2),(x2,y2)=(1,−3).

\begin{gathered}m = (-3-2)/(1-0)\\m = -5/1\\m = -5\end{gathered}

m=(−3−2)/(1−0)

m=−5/1

m=−5

Therefore, the slope of the graphed line is -5 having points (0,2) and (1,-3).

So, the -5 number best represents the slope of the graphed line.

Given: Mary, Gracie, and Nancy all have little collections of seashells, but Gracie has 5 more than 14 times as many as Mary does. So Mary has 16 shells

To find: Recognize the equation that captures this circumstance and recognize its resolution.

olution:

Let x be mary's shells, so according to question, we get:

gracie = 14x + 5   nancy  = 1 x+ 1

Now we have given that:

          gracie = nancy

So: 14x + 5 = 1 x + 1

  5/4x + 5 = 3/2x + 1

Fractions will be eliminated by multiplying by the common denominator of 4, but this step is optional; you can still work with fractions if you choose.

Now solving further, we get:

        5x + 20 = 6x + 4  5x - 6x = 4 - 20    -x = - 16    x = 16.

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We get the squares of the number as 1 , 4 , 9 , 16.......as given below, hence we get square of any number by product with itself.

The outcome of multiplying an integer by itself is referred to as the resultant number's square. In essence, a square number is a result of multiplying two identical numbers. The best illustration of a square number in geometry is the area of a square with side lengths of "n" (where n is an integer).

As an illustration, (-7)² = -7 -7 = 49 (two negative signs multiplied by each other result in a positive sign). The value of multiplying -7 and -7 is 49, which is a positive square number.

Below are the squares of 1 to 100.

1² = 1

2² = 4

3² = 9

4² = 16

5² = 25

6² = 36

7² = 49

8² = 64

9² = 81

10² = 100

11² = 121

12² = 144

13² = 169

14² = 196

15² = 225

16² = 256

17² = 289

18² = 324

19² = 361

20² = 400

21² = 441

22² = 484

23² = 529

24² = 576

25² = 625

26² = 676

27² = 729

28² = 784

29² = 841

30² = 900

31² = 961

32² = 1024

33² = 1089

34² = 1156

35² = 1225

36² = 1296

37² = 1369

38² = 1444

39² = 1521

40² = 1600

41² = 1681

42² = 1764

43² = 1849

44² = 1936

45² = 2025

46² = 2116

47² = 2209

4²2 = 2304

49² = 2401

50² = 2500

51² = 2601

52² = 2704

53² = 2809

54² = 2916

55² = 3025

56² = 3136

57² = 3249

58² = 3364

59² = 3481

60² = 3600

61² = 3721

62² = 3844

63² = 3969

64² = 4096

65² = 4225

66² = 4356

67² = 4489

68² = 4624

69² = 4761

70² = 4900

71² = 5041

72² = 5184

73² = 5329

74² = 5476

75² = 5625

76² = 5776

77² = 5929

78² = 6084

79² = 6241

80² = 6400

81² = 6561

82² = 6724

83² = 6889

84² = 7056

85² = 7225

86² = 7396

87² = 7569

88² = 7744

89² = 7921

90² = 8100

91² = 8281

92² = 8464

93² = 8649

94² = 8836

95² = 9025

96² = 9216

97² = 9409

98² = 9604

99² = 9801

100² = 10000

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Per year 40 squirrels are added

Any of the many statistical methods used to determine the value of a population's attribute based on observations of a sample taken from the population is known as an estimate. For instance, a point estimate is the one figure that is most likely to convey the property's value.

Estimates are created by multiplying the unit cost of the relevant goods by the amounts derived from the dimensions of the designs for various things needed to execute the project. Estimates' objectives: should be aware of the financial requirements for finishing the suggested task.

According to the given question

The birth rate is 0.06 and the death rate is 0.02 per person per year. The number of deaths per year for a population of 1000 is equal to 1000*0.02, or 20. The annual birth rate is equal to 1000 * 0.06, or 60. As a result, the population of squirrels grows by a total of 40 individuals each year

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7.8c + 6p - 3.4c - 10

combine c terms:

4.4c + 6p - 10

If you need to factor:

4.4c + 6p - 10

2(2.2c + 3p - 5)