Bert didn't finish 1/8 of the problems on a math test he made Mistakes on 1/6 of the problems the rest he answered correctly what fraction of the problems did he answer correctly

Answer:

P(y) take 36 h to do the job alone

Step-by-step explanation:

P(x) quantity of water pump by Pump X     and

P(y)  quantity of water pump by Pump Y

Then if  P(x)  pumped  1/3  of the water in a pool in 4 hours

Then in 1 hour P(x) will pump

    1/3    ⇒  4 h

    ? x    ⇒   1 h       x  =  1/3/4     ⇒   x  = 1/12

Then in 1 hour   P(x)  will pump  1/12 of the water of the pool

Now both pumps   P(x)  and P(y)   finished 2/3 of the water in the pool (left after the P(x) worked alone ) in 6 hours. Then

P(x)  +  P(y)  in    6 h    ⇒  2/3

                     in    1 h   ⇒    x ??      x  =  (2/3)/6     x  =  2/18    x  = 1/9

Then   P(x)  +  P(y)  pump   1/9 of the water of the pool in 1 h. We find out how long will take the two pumps to empty the pool

water in a pool is  9/9  ( the unit) then

  1  h    ⇒   1/9

   x ??  ⇒   9/9        x  = ( 9/9)/( 1/9)     ⇒   x  =  9 h

The two pumps would take 9 hours working together from the beggining

And in 1 hour  of work, both  pump 1/9 of the water, and P(x) pump 1/12 in 1 hour

Then in 1 hour P(y)

P(y)   =  1/9  -  1/12    ⇒    P(y)   = 3/108         P(y)   = 1/36

And to pump all the water  (36/36) P(y) will take

  1 h     1/36

  x ??   36/36         x  =  (36/36)/1/36

 x =  36 h

P(y) take 36 h to do the job alone

Answer:

The probability that sum of numbers rolled is a multiple of 3 or 4 is: [tex]\frac{7}{12}[/tex].

Step-by-step explanation:

The sample space for two fair die (dice) is given below:

[tex]\left[\begin{array}{ccccccc}&1&2&3&4&5&6\\1&2&3&4&5&6&7\\2&3&4&5&6&7&8\\3&4&5&6&7&8&9\\4&5&6&7&8&9&10\\5&6&7&8&9&10&11\\6&7&8&9&10&11&12\end{array}\right][/tex]

From the above table:

Number of occurrence where sum is multiple of 3 = 12

Number of occurrence where sum is multiple of 4 = 9

Total number in the sample space = 36

probability(sum is 3) = 12/36

probability(sum is 4) = 9/36

probability(sum is 3 or 4) [tex]=\frac{12}{36} +\frac{9}{36} \\=\frac{12 + 9}{36} \\=\frac{21}{36}\\=\frac{7}{12}[/tex]

Answer:

8 years

Step-by-step explanation:

Compound interest formula

[tex]A(t)= A_0(1+\frac{r}{n})^{nt}[/tex]

A(t) is the final amount 55000

A_0= 40000, r= 4% = 0.04, for quarterly n=4

[tex]55000=40000(1+\frac{0.04}{4})^{4t}[/tex]

divide both sides by 40000

[tex]1375=(1+\frac{0.04}{4})^{4t}[/tex]

[tex]1375=(1.01)^{4t}[/tex]

Take ln on both sides

[tex]ln(1375)=4tln(1.01)[/tex]

divide both sides by ln(1.01)

[tex]\frac{ln 1375}{ln 1.01}=4t[/tex]

Divide both sides by 4

t=8.00108

So it takes 8 years

The couple will need to invest their $40,000 at an interest rate of 4% compounded quarterly for about 7 years in order to reach their target of $55,000.

The subject of this question is compound interest. Compound interest is the interest computed on the initial principal as well as the accumulated interest from previous periods. Since the couple's money is being compounded quarterly, we will need to use this information in our calculations.

First, we must understand the compound interest formula which is:

A = P (1 + r/n)^(nt)

where,

Answer:

'x'  and 'y' have a positive association between them.

Step-by-step explanation:

We can see in the given data that as 'x' remains  constant or increases by 1 unit, or two units in it's two consecutive values, in each of the cases 'y' increases by 1 unit.

Hence, 'x'  and 'y' have a positive association between them.

Answer:

Step-by-step explanation:

Positive and negative association describe a relation between variables about a scatter plot.

A positive association happens when one variable increases while the other one also increases.

A negative association happens when one variable decreases while the other variable increases.

It's important to know that the variable that always increases is the independent variable, it increases no matter what.

In this case, we could say that we have a positive association, because while x increases, y also increases.

You could notice that there's some repetitive number. That's normal because a scatter plot doesn't describe a perfect linear relation at the beginning, actually, it's a bit messy, however, from such data we construct a linear relationship which is called "linear regression".

Therefore, the right answer is C.

Answer: The debt-to-equity ratio

Step-by-step explanation:

The debt-to-equity ratio is a company's debt as a percentage of its total market value. If your company has a debt-to-equity ratio of 50% or 70%, it means that you have $0.5 or $0.7 of debt for every $1 of equity

Answer:

Jasmine can load maximum of 25 boxes with herself on the elevator.

Step-by-step explanation:

Given:

Weight of Jasmine = 150 lb

Weight of each boxes  = 72 lb

Load elevator can carry = 2000 lb

we need to find the number of boxes that can be loaded

Let number of boxes be 'x'

Now we know that Maximum load the elevator can carry is 2000 lb.

So We can say Weight of jasmine plus Number of boxes multiplied by Weight of each boxes should be less than or equal to Load elevator can carry.

Framing in equation form we get;

[tex]150+72x\leq 2000[/tex]

Solving the equation we get:

We will first Subtract 150 on both side;

[tex]150+72x-150\leq 2000-150\\\\72x\leq 1850[/tex]

Now Dividing both side by 72 by using Division property we get;

[tex]\frac{72x}{72}\leq \frac{1850}{72}\\\\x\leq 25.69[/tex]

Hence Jasmine can load maximum of 25 boxes with herself on the elevator.

Answer:

  y = 2x +7

Step-by-step explanation:

We are given two points on the growth curve: (weeks, ounces) = (2, 11) or (4, 15).

These can be used to write the equation of a line using the 2-point form:

  y = (y2 -y1)/(x2 -1x)(x -x1) +y1

  y = (15 -11)/(4 -2)(x -2) +11

  y = 4/2(x -2) +11

  y = 2x +7 . . . . . y = weight in ounces x weeks after birth

_____

Comment on the problem

There are an infinite number of equations that can be written to go through the two given points. A linear equation is only one of them.

The 95% confidence interval for the true mean score of all teenage boys is approximately 76,080 to 83,920.

To calculate the 95% confidence interval for the true mean score of all teenage boys, we'll use the formula for the confidence interval:

[tex]\[ \text{Confidence Interval} = \bar{x} \pm Z \left( \frac{\sigma}{\sqrt{n}} \right) \][/tex]

where:

- [tex]\(\bar{x}\)[/tex] is the sample mean,

- Z is the Z-score corresponding to the desired confidence level,

- [tex]\(\sigma\)[/tex] is the population standard deviation, and

- n is the sample size.

For a 95% confidence interval, the Z-score is approximately 1.96.

Given:

- Sample mean [tex](\(\bar{x}\))[/tex] = 80,000

- Population standard deviation (\(\sigma\)) = 20,000

- Sample size (n) = 100

[tex]\[ \text{Confidence Interval} = 80,000 \pm 1.96 \left( \frac{20,000}{\sqrt{100}} \right) \][/tex]

Calculate the standard error (SE):

[tex]\[ SE = \frac{\sigma}{\sqrt{n}} = \frac{20,000}{\sqrt{100}} = 2,000 \][/tex]

Now substitute the values into the formula:

[tex]\[ \text{Confidence Interval} = 80,000 \pm 1.96 \times 2,000 \][/tex]

[tex]\[ \text{Confidence Interval} = 80,000 \pm 3,920 \][/tex]

The 95% confidence interval is from 80,000 - 3,920 to 80,000 + 3,920.

Answer:

r  = 225 Mil/h     speed of the airplane in still air

Step-by-step explanation:

Then:

d  is traveled distance   and r  the speed of the airplane in still air

so the first equation is for a 4 hours trip

as  d = v*t

d  =  4 *  ( r  + 25)    (1)          the speed of tail wind  (25 mil/h)

Second equation the trip back in 5 hours

d  =  5  * ( r  - 25 )    (2)

So we got a system of two equation and two unknown variables  d  and

r

We solve it by subtitution

from equation (1)     d  =  4r  + 100

plugging in equation 2

4r  + 100  = 5r -  125    ⇒   -r  =  -225     ⇒    r  = 225 Mil/h

And distance is :

d  =  4*r  +  100         ⇒ d  =  4 * ( 225)  +  100

d  =  900 +   100

d  = 1000 miles

To find all x that satisfy the inequality (2x + 10)(x + 3) < (3x + 9)(x + 8), we expand and simplify the inequality, solve for x by considering the sign of each factor, and express the answer in interval notation. The solution is (-7, -6).

To find all x that satisfy the inequality (2x + 10)(x + 3) < (3x + 9)(x + 8), we will first expand and simplify the inequality. Then, we will solve for x by considering the sign of each factor.

The interval notation for the solution is (-7, -6). This means that all values of x between -7 and -6 (exclusive) satisfy the inequality.

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Answer:

C. 4

Step-by-step explanation:

f(c) = 28 = 2x² - 4

28 = 2x² - 4

32 = 2x²

16 = x²

±4 = x

x = 4

c = 4

Answer:

Option c) is correct

ie., c=4 represents the value of c suchthat function  [tex]f(c)=28[/tex]

Step-by-step explanation:

Given function f is defined by [tex]f(x)=2x^{2}-4[/tex]

To find the value of "c" such that  [tex]f(c)=28[/tex]

Therefore put x=c in the given function as

[tex]f(x)=2x^{2}-4[/tex]

[tex]f(c)=2c^{2}-4[/tex]

and we have [tex]f(c)=28[/tex]

Now equating the two functions

[tex]f(c)=2c^{2}-4=28[/tex]

[tex]2c^{2}-4=28[/tex]

[tex]2c^{2}=28+4[/tex]

[tex]c^{2}=\frac{32}{2}[/tex]

[tex]c^{2}=16[/tex]

[tex]c=4[/tex]

Therefore [tex]c=4[/tex]

Option c) is correct

ie., c=4 represents the value of c suchthat function  [tex]f(c)=28[/tex]

Answer:

YES

NO

NO

Step-by-step explanation:

The given polynomial is: [tex]$ f(x) =  x^3 + 4x^2 - 25x - 100 $[/tex]

(x - a) is a factor of a polynomial iff x = a is a solution to the polynomial.

To check if (x - 5) is a factor of the polynomial f(x), we substitute x = 5 and check if it satisfies the equation.

∴ f(5) = 5³ + 4(5)² - 25(5) - 100

        = 125 + 100 - 125 - 100

        = 225 - 225

        = 0

We see, x = 5 satisfies f(x). So, (x - 5) is a factor to the polynomial.

Now, to check (x + 2) is a factor.

i.e., to check x = - 2 satisfies f(x) or not.

f(-2) = (-2)³ + 4(-2)² - 25(-2) - 100

      = -8 + 16 + 50 - 100

     = -108 + 66

     ≠ 0

Therefore, (x  + 2) is not a factor of f(x).

To check (x - 4) is a factor.

∴ f(4) = 4³ + 4(4)² - 25(4) - 100

        = 64 + 64 - 100 - 100

        = 128 - 200

        ≠ 0

Therefore, (x - 4) is not a factor of f(x).

Answer:

The answers are (x-5) YES (x+2) NO (x-4) NO

Step-by-step explanation:

I took the test :)

Answer:

50 grams

Step-by-step explanation:

Let the amount of cheese required by the recipe be "x"

Ann increased 60% from original amount and then used up 80 grams. Thus:

Original, increased by 60%, became 80

This translated to algebraic equation would be:

x + 0.6x = 80

Note:  60% = 60/100 = 0.6

So we can solve the above equation for "x" and get our answer. Shown below:

[tex]x + 0.6x = 80\\1.6x=80\\x=\frac{80}{1.6}\\x=50[/tex]

Hence,

the recipe required 50 grams of cheese

Answer:the cost of each CD for a member is $5.25

Step-by-step explanation:

The membership costs $22 per year, and then a member can buy CDs at a reduced price.

Let x represent the cost of each CD for a member.

Let n represent the number of CDs that each member buys. This means that the total cost of buying n CDs for a member for a year would be

22 + nx

If a member buys 17 CDs in one year, the cost is $111.25. This means that

17x + 22 = 111.25

17x = 111.25 - 22 = 89.25

x = 89.25/175 = 5.25

Answer:

25 miles

Step-by-step explanation:

Given: A boat sail 20 miles west of the port and then 15 miles south to an island.

Picture attached.

The distance from port to island could be measured in a straight line. It will form a hypotenous.

∴ we can use Pythogorean theorem to find the distance.

[tex]h^{2} = a^{2} +b^{2}[/tex]

Where, "a" is adjacent= 20 miles and "b" is opposite= 15 miles.

[tex]h^{2} = 20^{2} +15^{2}[/tex]

⇒ [tex]h^{2} = 400+225= 625[/tex]

⇒[tex]h^{2} = 625[/tex]

⇒[tex]h= \sqrt{625}= \sqrt{25^{2} }[/tex]

We know [tex]\sqrt{x^{2} } = x[/tex].

∴[tex]h= 25\ miles[/tex]

∴ Distance of Port from the Island is 25 miles.

Final answer:

Using the Pythagorean theorem, the straight-line distance from the port to the boat, after traveling 20 miles west and 15 miles south, is found to be 25 miles.

Explanation:

The problem presented involves calculating the straight-line distance, or 'hypotenuse', of a right-angled triangle formed by the boat's journey from the port, 20 miles west and then 15 miles south. We will use the Pythagorean theorem to solve this problem. The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). The formula is c² = a² + b².

Let's apply this formula to find the distance:

The distance travelled west (a) is 20 miles.

The distance travelled south (b) is 15 miles.

We can calculate the straight-line distance using the formula:

c² = 20² + 15²
c² = 400 + 225
c² = 625
c = √625
c = 25 miles

Therefore, the boat is 25 miles away from the port if measured in a straight line.

Answer:

4

Step-by-step explanation:

To restore her server after a failure on Wednesday morning, Nancy would need to restore the full backup from Sunday, and then restore the differential backup from Tuesday.

In Nancy's case, she would need two backups to fully restore her server. These would be the full backup from Sunday and the differential backup from Tuesday. Here's why:

Because Nancy performs full backups every Sunday, the full backup will have all the data up until Sunday at 1 A.M. The differential backup from Tuesday will contain all the changes that occurred on Monday and Tuesday until 1 A.M.

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Answer:

x = π/6, π/2, 5π/6, 3π/2

Step-by-step explanation:

cos x = sin(2x)

Use double angle formula.

cos x = 2 sin x cos x

Move everything to one side and factor.

cos x − 2 sin x cos x = 0

cos x (1 − 2 sin x) = 0

Set each factor to 0 and solve.

cos x = 0

x = π/2, 3π/2

1 − 2 sin x = 0

sin x = 1/2

x = π/6, 5π/6

The total solution is:

x = π/6, π/2, 5π/6, 3π/2

Final answer:

The solutions to the equation cos x = sin 2x in the interval [0, 2π) are π/6, π/2, 5π/6, and 3π/2, derived by using the identity sin 2x = 2 sin x cos x and considering cases for cos x = 0.

Explanation:

To find all solutions to the equation cos x = sin 2x in the interval [0, 2π), we first need to use a trigonometric identity to express both sides of the equation with either sine or cosine. The identity sin 2x = 2 sin x cos x can be used here. Substituting it into our original equation, we get:

cos x = 2 sin x cos x

To solve this equation, we can divide both sides by cos x, given that cos x ≠ 0:

1 = 2 sin x

sin x = 1/2

Using the unit circle or trigonometric tables, we know that sin x takes the value of 1/2 at x = π/6 and x = 5π/6 in the interval [0, 2π). Additionally, we must consider the case when cos x = 0 to avoid division by zero. This occurs at x = π/2 and x = 3π/2, which are also solutions to the original equation given that sin(2(π/2)) = sin(π) = 0 and sin(2(3π/2)) = sin(3π) = 0, which are equal to cos(π/2) and cos(3π/2) respectively. Thus, the complete set of solutions in the interval [0, 2π) is π/6, 5π/6, π/2, and 3π/2.

Final answer:

Ricky's best time running 200 meters is 23.34 seconds, and his worst time is 23.45 seconds. The difference between his best and worst times is 0.11 seconds.

Explanation:

The question asks what the difference is between Ricky's best and worst times running 200 meters. To find this, we need to identify the fastest (lowest) time and the slowest (highest) time from the provided times: 23.37 sec, 23.45 sec, 23.44 sec, and 23.34 sec. Ricky's best time is 23.34 seconds, and his worst time is 23.45 seconds. The difference between these times is calculated by subtracting the best time from the worst time.

Worst time (slowest): 23.45 sec

Best time (fastest): 23.34 sec

Difference: 23.45 sec - 23.34 sec = 0.11 sec

Therefore, the difference between Ricky's best and worst time is 0.11 seconds.

Answer:

  3√29 ≈ 16.155

Step-by-step explanation:

The distance formula can be used to find the length of the given side.

  d = √((x2-x1)^2 +(y2-y1)^2)

  PQ = √((3-1)^2 +(10-5)^2) = √(4 +25) = √29

The equilateral triangle has three same-length sides (the literal meaning of "equilateral"), so the perimeter is ...

  Perimeter = 3×PQ = 3√29

Area of path and garden = 11 x 11 = 121 square feet.

Area of garden only = 7 x 7 = 49 square feet.

Area of just the path = 121 - 49 = 72 square feet.

To solve for the width of the rectangle, use the equation width = (2L - 1) / 2. Set up the equation L * width = 91 and simplify it to 2L² - L - 182 = 0. Solve the quadratic equation to find the possible values for L, which will give us the width of the rectangle.

To solve for the width of the rectangle, we can use the equation width = (2L - 1) / 2, where L is the length of the rectangle. Given that the area of the rectangle is 91 square feet, we can set up the equation as L * width = 91. Substituting the first equation into the second equation, we get:

L * ((2L - 1) / 2) = 91

Simplifying the equation, we have:

2L² - L - 182 = 0

We can then solve this quadratic equation to find the possible values for L, which will give us the width of the rectangle.

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The correct equation to solve for the width of the rectangle with an area of 91 square feet and a length that is one foot less than twice its width is 2x^2 - x - 91 = 0. Therefore, the correct answer is option C.

The student's question involves finding the width of a rectangle, given that the length is one foot less than twice its width and that the area of the rectangle is 91 square feet.

We can represent the width of the rectangle with x and the length as 2x - 1. The area of a rectangle is found by multiplying its length by its width, which gives us the equation x(2x - 1) = 91.

Expanding this equation, we get 2x^2 - x - 91 = 0. Therefore, the correct equation to solve for the width of the rectangle is 2x^2 - x - 91 = 0.

The probable question may be:

A rectangle has a length that is one foot less than twice its width. it the area of the rectangle is 91 square feet then which of the following equations could be used to solve for its the width of the rectangle?

O x² + 2x + 91 = 0
O 2x² + x + 91 = 0
O 2x² - x - 91 = 0

O x² - 2x - 91 = 0

The expression which  gives the area of the RED rectangle is (A+B)(C+D)-C(A+B)-BD.  option C is correct.

The area of rectangle is obtained by multiplying the length and width.

The length of the rectangle is C+D

The width of the rectangle is A+B.

Now the complete area of rectangle :

Area = (A+B)(C+D)

Now to find area of rectangle which is red we have to subtract the red rectangle which are in blue:

The area of left side rectangle which is blue:

Area =C(A+B)

The area of rectangle below red rectangle:

Area =BD

So the area of red rectangle : (A+B)(C+D)-C(A+B)-BD.

Hence, option C is correct, the expression which  gives the area of the RED rectangle is (A+B)(C+D)-C(A+B)-BD.

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Yes because the number say on one side

yes because the numbers stay on one side of the number line

Answer:

Step-by-step explanation:

Looking at both triangles, angle P in triangle PQR = 38 degrees. Angle N in triangle LMN = 38 degrees. Both angles are equal.

Side PQ in triangle PQR = 16

Side MN in triangle LMN = 8

Therefore,

PQ/MN = 16/8 = 2

Side PR in triangle PQR = 14

Side LN in triangle LMN = 7

Therefore,

PR/LN = 14/7 = 2

Therefore, triangle PQR is similar to

triangle LMN because

1) the length of PQ is proportional to the length of MN.

2) the length of PR is proportional to the length of LN

3) angle P = angle N

4) Therefore, QR is also proportional to ML

Therefore,

PQ/MN = PR/LN = QR/ML = 2

Answer:

5.30 PM

Step-by-step explanation:

time taken=(1/2)×(1/35)×245=7/2 hrs=3hrs 30 min.

so time =2+3.30=5.30 PM

Answer: 5:30pm

Step-by-step explanation:

Distance of family destination from the starting point = 245miles

Average speed = 35miles/1/2hour

Therefore 245 miles = 245/35 x 30mins.

7 x 30 = 210minutes.

Converted to hour by dividing by 60

210/60 = 3hours 30minutes.

Current local time at the point of commencement

Therefore, the arrival time at their destination = 14hours + 3hours 30mins.

= 17hours 30minutes. Convert to local time

17hours 30minutes = 12hours

= 5:30 pm in the evening

They have succeeded in spending 3hours 30minutes on the road. The final answer = 5:30pm.

Answer:

[tex]\frac{15}{32}[/tex]

Step-by-step explanation:

Given that a coin is flipped 5 times. Let us assume it is a fair coin with probability for head or tail equally likely and hence 0.50

Sample space would have [tex]2^5 = 32[/tex] possibilities

For two heads never occurring in a row favourable outcomes are

HTTTT,HTTHT,HTTTH,  HTHTH,  HTHTT, THTHT, THTTT, THTTH, TTHTH, TTHTT, TTTHT, TTTTH, TTTTT, TTTTH

Hence probability =

Favorable outcomes/Total outcomes=

[tex]\frac{15}{32}[/tex]

Answer: [tex]\dfrac{1}{8}[/tex]

Step-by-step explanation:

The total outcomes of tossing a coin = 2

The total number of possible outcomes of flipping coin three times =2 x 2 x 2 = 8

Favorable outcome= (Head, tail, Head) in order

i.e. Number of Favorable outcomes = 1

We know that , Probability = [tex]\dfrac{\text{Favorable outcomes}}{\text{Total outcomes}}[/tex]

Therefore , The required probability [tex]=\dfrac{1}{8}[/tex]

Hence , the probability she gets (Head, tail, Head) in that order[tex]=\dfrac{1}{8}[/tex]

Answer: A. See photo for work.

Step-by-step explanation:

Answer:the cost of one burger is $1.4 and the cost of one fry is $1.6

Step-by-step explanation:

Let x represent the cost of one burger.

Let y represent the cost of one fry.

A man buys 3 burgers and 2 jumbo deluxe fries for $7.40. This means that

3x + 2y = 7.4 - - - - - - - - - 1

A woman buys one burger and 4 jumbo deluxe fries for $7.80. It means that

x + 4y = 7.8 - - - - - - - - -2

Multiplying equation 1 by 1 and equation 2 by 3, it becomes

3x + 2y = 7.4

3x + 12y = 23.4

Subtracting

- 10y = - 16

y = - 16/- 10 = 1.6

Substituting y = 1.6 into equation 2, it becomes

x + 4 × 1.6 = 7.8

x = 7.8 - 6.4 = 1.4

Answer:

The answer to your question is letter C

Step-by-step explanation:

Process

1.- Find two points of the dotted line

   A (0, 4)

   B (3, 5)

2.- Find the slope of the line

   [tex]m = \frac{y2 - y1}{x2 - x1}[/tex]

Substitution

   [tex]m = \frac{5 - 4}{3 - 0}[/tex]

   [tex]m = \frac{1}{3}[/tex]

3.- Write the equation of the line

   y - y1 = m(x - x1)

   y - 4 = 1/3(x - 0)

   y - 4 = 1/3x

   y = 1/3x + 4

4.- Write the inequality

    We are interested on the upper area so

    y > 1/3x + 4

Answer: C) y > 1/3x + 4

Step-by-step explanation: The line is dashed hence >. The slope is 1/3 and the y- intercept is 4. This makes the inequality y > 1/3x + 4.

Hope this helps!  :)

Answer:

The number of  people that received copies of the letter at the twentieth stage is 9.537 × 10¹³ .

Step-by-step explanation:

Using the discrete model,

a_k = r a_(k-1) for all integers k ≥ 1 and a₀ = a

then,

aₙ = a rⁿ for all integers n ≥ 0

Let a_k be equal to the number of people who receive a copy of the chain letter at a stage k.

Initially, one person has the chain letter (which the person will send to five other people at stage 1). Thus,

a = a₀ = 1

The people who received he chain letter at stage (k - 1),  will send a letter to five people at stage k and thus per person at stage (k - 1), five people will receive the letter. Therefore,

a_k = 5 a_(k - 1)

Thus,

aₙ = a rⁿ = 1 · 5ⁿ = 5ⁿ

The number of  people that received copies of the letter at the twentieth stage is

a₂₀ = (5)²⁰ = 9.537 × 10¹³ copies