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4 months ago

The sequence 6, 18, 54, 162, … shows the number of pushups Kendall did each week, starting

Mathematics

1 answer:

Svet_ta [12.7K]4 months ago

5 0

a) The sequence is geometric.

b) The subsequent term is 486

c) The rule is: $a_n=6(3)^{n-1}$

The 20th term will be: a_{20}=6973568802

Step-by-step explanation:

The series 6, 18, 54, 162, ... depicts the number of push-ups Kendall performed each week, starting from her initial week of exercise.

Part a) Is this sequence arithmetic or geometric?

An arithmetic sequence has a constant difference between consecutive terms.

In the given series 6, 18, 54, 162,

18-6 = 12

54-18= 36

The differences are not constant, thus it isn’t arithmetic.

A geometric sequence maintains a constant ratio between consecutive terms.

In this series 6, 18, 54, 162,

18/6 = 3

54/18 = 3

162/54 = 3

The ratios remain constant; therefore, the sequence is geometric.

The sequence in question is geometric.

Part b) How can you determine the next number in the series?

The next number can be calculated using the following formula:

$a_n=a(r)^{n-1}$

where a_n= nth term

a= first term

r= common ratio

The next number represents the 5th term of the sequence.

To find the 5th term:

$a_n=a(r)^{n-1}\\a_5=6(3)^{5-1}\\a_5=6(3)^4\\a_5=6*81\\a_5=486$

Thus, the next term in the sequence is 486

Part c) What is the rule for determining the 20th week? Calculate the 20th week.

The rule for determining the 20th week is:

$a_n=6(3)^{n-1}$

Here n=20

Calculating the 20th week

$a_n=6(3)^{n-1}\\a_{20}=6(3)^{20-1}\\a_(20)=6(3)^{19}\\a_{20}=6973568802$

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Three people get into an empty elevator at the first floor of a building that has 10 floors. Each presses the button for their d

Zina [12379]

Answer:

The chance of the buttons for three consecutive floors being pressed is $\frac{1}{12}$ or 0.0833

Step-by-step explanation:

Consider the information given.

The users have an equal probability of selecting floors 2 through 10.

Thus, the quantity of buttons is 9.

Each person will press the button for their desired floor (unless someone else has already selected that floor).

This implies that if multiple individuals aim for the same floor, only one will push the button.

Therefore, the total arrangements are calculated as: 9 × 8 × 7 = 504.

Choosing three consecutive floors means they can be:

2,3,4 or 3,4,5 or 4,5,6 or 5,6,7 or 6,7,8 or 7,8,9 or 8,9,10.

With three individuals, they can be arranged in 3! different ways.

As a result, the total number of favorable outcomes in this scenario is 7 × 3! = 42.

$Probability = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$

$Probability = \frac{42}{504}=\frac{1}{12}$

Hence, the probability that the buttons for 3 consecutive floors get pressed is $\frac{1}{12}$ or 0.0833

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3 months ago