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

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 $e^1$?

In an arithmetic sequence $a_n$, the initial value is $a_0=4$ and the common difference is $d=-3$. What is the value of $a_5$?

An arithmetic sequence is given by $a_n=7-2n$ for $n\ge 0$. A linear function $f$ of the form $f\left(x\right)=b+mx$ has the same initial value and the same constant rate of change as the sequence. Which of the following is an expression for $f\left(x\right)$?

A geometric sequence is given by $g_n=5·{\left(\frac{1}{3}\right)}^n$ for $n\ge 0$. An exponential function $f\left(x\right)=a{b}^x$ models the sequence such that $f\left(n\right)=g_n$ for all whole numbers $n$. What are the values of $a$ and $b$?

The function $f$ is given by $f\left(x\right)=4·{49}^x$. Which of the following is an equivalent form for $f\left(x\right)$ ?

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 $x$.

Which of the following statements is true?

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?

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?

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

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

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 $x$.

Which of the following statements is true?

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

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

The function $f$ is given by $f\left(x\right)=30·e^{0.4x}$. The function $f$ can also be written in the form $f\left(x\right)=30·b^x$, where $b$ is a positive constant. What is the value of $b$?

The terms of an arithmetic sequence are given by $a_n$. If $a_2=11$ and $a_6=27$, what is the value of $a_{10}$?

The function $h$ is given by $h\left(x\right)=2·5^{2x-1}$. Which of the following is an equivalent form for $h\left(x\right)$?

The terms of an arithmetic sequence are given by $a_n$, where $a_0=100$ and the common difference is $8$. The terms of a geometric sequence are given by $g_n$, where $g_0=100$ and the common ratio is $1.05$. Which of the following statements is true?

The exponential function $f$ is defined by $f\left(x\right)=a·b^x$, where $a$ and $b$ are positive constants. The graph of $f$ in the $xy$-plane passes through the points $\left(1,12\right)$ and $\left(6,96\right)$. What is the value of $f\left(0\right)$?

The expressions $x$, $x+6$, and $4x+9$ are the first three terms of a geometric sequence, in that order. What is the sum of all possible values of $x$?

The terms $a_1$, $a_2$, $a_3$ of an arithmetic sequence have a positive common difference. Their sum is $27$ and their product is $693$. What is the value of the common difference?