AP®MathsCollege BoardPrecalculusExam QuestionsExponential & Logarithmic FunctionsSequences & Exponential Functions

Sequences & Exponential Functions (College Board AP® Precalculus): Exam Questions

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Sequences & Exponential Functions

1
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1 mark

The general term of a geometric sequence is given by g subscript n equals 3 times 2 to the power of n, for n \geq 0. What is the value of g subscript 3?

  • 18

  • 24

  • 54

  • 216

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2
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1 mark

The natural base e is often used in exponential functions to model contextual scenarios. Which of the following is the best approximation for the value of e^{1}?

  • 1 . 000

  • 2 . 718

  • 3 . 142

  • 10 . 000

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3
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1 mark

In an arithmetic sequence a subscript n, the initial value is a_{0} = 4 and the common difference is d = - 3. What is the value of a subscript 5?

  • negative 972

  • negative 11

  • negative 8

  • 19

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4
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1 mark

An arithmetic sequence is given by a subscript n equals 7 minus 2 n for n \geq 0. A linear function space f of the form space f open parentheses x close parentheses equals b plus m x has the same initial value and the same constant rate of change as the sequence. Which of the following is an expression for space f open parentheses x close parentheses?

  • space f open parentheses x close parentheses equals negative 2 plus 7 x

  • space f open parentheses x close parentheses equals 2 minus 7 x

  • space f open parentheses x close parentheses equals 7 minus 2 x

  • space f open parentheses x close parentheses equals 7 plus 2 x

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5
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1 mark

A geometric sequence is given by g subscript n equals 5 times open parentheses 1 third close parentheses to the power of n for n greater or equal than 0. An exponential function space f open parentheses x close parentheses equals a b to the power of x models the sequence such that space f open parentheses n close parentheses equals g subscript n for all whole numbers n. What are the values of a and b?

  • a equals 1 third and b = 5

  • a equals 5 over 3 and b equals 1 third

  • a = 5 and b equals negative 1 third

  • a = 5 and b equals 1 third

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6
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1 mark

The function space f is given by space f open parentheses x close parentheses equals 4 times 49 to the power of x. Which of the following is an equivalent form for space f open parentheses x close parentheses ?

  • space f open parentheses x close parentheses equals 2 times 7 to the power of 2 x end exponent

  • space f open parentheses x close parentheses equals 4 times 7 to the power of x divided by 2 end exponent

  • space f open parentheses x close parentheses equals 2 times 7 to the power of x divided by 2 end exponent

  • space f open parentheses x close parentheses equals 4 times 7 to the power of 2 x end exponent

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7
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1 mark

The exponential function f is defined by f \left(x\right) = a b^{x}, where a and b are positive constants. The table gives values of f \left(x\right) at selected values of x.

x

0

1

2

3

4

f \left(x\right)

48

24

12

6

3

Which of the following statements is true?

  • f demonstrates exponential decay because a > 0 and 0 < b < 1.

  • f demonstrates exponential decay because a > 0 and b > 1.

  • f demonstrates exponential growth because a > 0 and 0 < b < 1.

  • f demonstrates exponential growth because a > 0 and b > 1.

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8
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1 mark

In a certain experiment, the number of cells in a culture can be modeled using a geometric sequence, where the first day of the experiment is day 1. The number of cells on day 3 was 1,500, and the number of cells on day 9 was 96,000. What was the number of cells on day 6 based on the model?

  • 48{,}750

  • 12{,}000

  • 6{,}000

  • 375

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9
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1 mark

In a nature reserve, the number of rabbits can be modeled using a geometric sequence, where the first month of the study is month 1. The number of rabbits counted in month 2 was 200, and the number counted in month 8 was 12,800. What was the number of rabbits in month 5, based on the model?

  • 25

  • 800

  • 1 , 600

  • 6 , 500

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10
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1 mark
A scatter graph with the horizontal axis labelled n (from 0 to 8, tick marks at each integer) and the vertical axis labelled gₙ (from 0 to 30). Six discrete points are plotted — not connected — at approximately (1, 27), (2, 9), (3, 3), (4, 1), (5, ⅓), and (6, ⅑). The points decrease rapidly from left to right, consistent with a geometric sequence with common ratio ⅓.

Values of the terms of a geometric sequence g_n are graphed in the figure. Which of the following is an expression for the nth term of the geometric sequence?

  • g_n = 27\left(\frac{1}{3}\right)^{n-1}

  • g_n = 27(3)^{n-1}

  • g_n = 27\left(\frac{1}{3}\right)^{n}

  • g_n = 81\left(\frac{1}{3}\right)^{n-1}

Choose your answer

11
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1 mark

The function f is given by f\left(x\right) = 3 \cdot 8^{x}. Which of the following is an equivalent form for f\left(x\right)?

  • f\left(x\right) = 3 \cdot 2^{x/3}

  • f\left(x\right) = 6 \cdot 2^{3x}

  • f\left(x\right) = 6 \cdot 2^{x/3}

  • f\left(x\right) = 3 \cdot 2^{3x}

Choose your answer

12
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1 mark

The exponential function f is defined by f\left(x\right) = ab^{x}, where a and b are positive constants. The table gives values of f\left(x\right) at selected values of x.

x

0

1

2

3

4

f\left(x\right)

5

15

45

135

405

Which of the following statements is true?

  • f demonstrates exponential decay because a > 0 and 0 < b < 1.

  • f demonstrates exponential decay because a > 0 and b > 1.

  • f demonstrates exponential growth because a > 0 and 0 < b < 1.

  • f demonstrates exponential growth because a > 0 and b > 1.

Choose your answer

13
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1 mark

The table gives values of a function space f at selected values of x.

x

2

4

6

8

space f open parentheses x close parentheses

27

45

75

125

Which of the following statements is true about the function space f?

  • space f is best modeled by a linear function because the output values change at a constant rate over equal-length input-value intervals.

  • space f is best modeled by a linear function because the output values change proportionally over equal-length input-value intervals.

  • space f is best modeled by an exponential function because the output values change at a constant rate over equal-length input-value intervals.

  • space f is best modeled by an exponential function because the output values change proportionally over equal-length input-value intervals.

Choose your answer

14
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1 mark

The function space f is given by space f open parentheses x close parentheses equals 30 times e to the power of 0.4 x end exponent. The function space f can also be written in the form space f open parentheses x close parentheses equals 30 times b to the power of x, where b is a positive constant. What is the value of b?

  • 0 . 400

  • 1 . 087

  • 1 . 492

  • 2 . 718

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15
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1 mark

The terms of an arithmetic sequence are given by a subscript n. If a subscript 2 equals 11 and a subscript 6 equals 27, what is the value of a_{10}?

  • 39

  • 43

  • 47

  • 67

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16
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1 mark

The function h is given by h open parentheses x close parentheses equals 2 times 5 to the power of 2 x minus 1 end exponent. Which of the following is an equivalent form for h open parentheses x close parentheses?

  • 2 over 5 times 10 to the power of x

  • 2 over 5 times 25 to the power of x

  • 2 times 25 to the power of x

  • 10 times 25 to the power of x

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17
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1 mark

The terms of an arithmetic sequence are given by a subscript n, where a subscript 0 equals 100 and the common difference is 8. The terms of a geometric sequence are given by g subscript n, where g subscript 0 equals 100 and the common ratio is 1.05. Which of the following statements is true?

  • g subscript n less than a subscript n for all whole numbers n greater or equal than 1.

  • g subscript n greater than a subscript n for all whole numbers n greater or equal than 1.

  • The values of a subscript n are initially greater than the values of g subscript n, but the values of g subscript n eventually exceed the values of a subscript n.

  • The values of g subscript n are initially greater than the values of a subscript n, but the values of a subscript n eventually exceed the values of g subscript n.

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18
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1 mark

The exponential function space f is defined by space f open parentheses x close parentheses equals a times b to the power of x, where a and b are positive constants. The graph of space f in the x y-plane passes through the points open parentheses 1 comma 12 close parentheses and open parentheses 6 comma 96 close parentheses. What is the value of space f open parentheses 0 close parentheses?

  • 6 . 000

  • 7 . 500

  • 7 . 917

  • 12 . 000

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