Question

1. Write the recursive formula and the explicit formula for the sequence {3,6,12,24,48,...}. Show all work.

2. In a paragraph, explain whether or not all geometric sequences are exponential functions.

3. Write a real world situation for the function f(x) = 3(1.06x) + 2. Be sure to identify all key features of the function as what they represent in your situation.

4. In a paragraph, compare and contrast constant rate of change and average rate of change.

5. What is the average rate of change of the function g(x) = 3(2x) - 6 over the interval 0<=x<=3? Show all work.

Answer 1

1. The given sequence is a geometric sequence.

The nth term of a geometric sequence is given by

where is the nth term of the sequence,

is the first term of the sequence and

r is the common ratio.

In the given sequence,

a₁ = 3 and r = 2

∴ aₙ = 3(2ⁿ⁻¹)

2. All geometric sequences are discrete form of exponential functions.

A geometric sequence is given by

aₙ = a₀ rⁿ , where n is a discrete value, in other words, a counting integer.

If the discrete value, n is replaced by a continuous variable, x , the resulting function is exponential in nature.

3. Here, the given function is f(x) = 3(1.06 x) + 2.

We see that, f(x) increases with increase in x.

Let us assume x to be the amount of gold in grams and f(x) to be the price of gold.

The cost of gold is directly proportional to the amount of gold bought and there is a flat tax collected, which is 2 bucks.

4. Constant rate of change : When a quantity changes uniformly over the another quantity, it is said to constant change over the other. For example, when a car covers a distance of 50 km every hour, the car is said to have a constant speed, which is rate of change of distance.

Average rate of change : It is the average rate at which one quantity is changing with respect to another. Suppose let us take the car to be covering a distance of 50 km in the first hour and 60 km in the second hour. Here, the rate of change in first hour is 50 kmph, where as in the second is 60 kmph. The average rate of change is 55 kmph.

5. Average rate of change is the slope of the function.

Here the function is g(x)=6x-6

If y = mx+c, the slope is m.

Here, the function is also of the same form, where m =6 and c = -6.

So, the slope of the equation is 6.

Average rate of change is 6 in the interval, 0≤x≤3

Answer:

Step-by-step explanation:

Answer 2

1. The given sequence is a geometric sequence.

The nth term of a geometric sequence is given by

`a_(n) = a_(1) r^(n-1)`

where
`a_(n)` is the nth term of the sequence,

`a_(1)` is the first term of the sequence and

r is the common ratio.

In the given sequence,

a₁ = 3 and r = 2

∴ aₙ = 3(2ⁿ⁻¹)

2. All geometric sequences are discrete form of exponential functions.

A geometric sequence is given by

aₙ = a₀ rⁿ , where n is a discrete value, in other words, a counting integer.

If the discrete value, n is replaced by a continuous variable, x , the resulting function is exponential in nature.

3. Here, the given function is f(x) = 3(1.06 x) + 2.

We see that, f(x) increases with increase in x.

Let us assume x to be the amount of gold in grams and f(x) to be the price of gold.

The cost of gold is directly proportional to the amount of gold bought and there is a flat tax collected, which is 2 bucks.

4. Constant rate of change : When a quantity changes uniformly over the another quantity, it is said to constant change over the other. For example, when a car covers a distance of 50 km every hour, the car is said to have a constant speed, which is rate of change of distance.

Average rate of change : It is the average rate at which one quantity is changing with respect to another. Suppose let us take the car to be covering a distance of 50 km in the first hour and 60 km in the second hour. Here, the rate of change in first hour is 50 kmph, where as in the second is 60 kmph. The average rate of change is 55 kmph.

5. Average rate of change is the slope of the function.

Here the function is g(x)=6x-6

If y = mx+c, the slope is m.

Here, the function is also of the same form, where m =6 and c = -6.

So, the slope of the equation is 6.

Average rate of change is 6 in the interval, 0≤x≤3