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

The general term of a geometric sequence is given by $g_n=4·{\left(-3\right)}^n$ for $n\ge 0$.

Determine the exact value of $g_3$.

The function $f$ is given by $f\left(x\right)=3^{x+2}$.

Rewrite $f\left(x\right)$ as an equivalent expression of the form $a·3^x$, where $a$ is a constant. Identify the exact value of $a$.

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

Based on the table, which of the following function types best models function $f$: linear, quadratic, exponential, or logarithmic?

The first three terms of an arithmetic sequence are $5$, $12$, and $19$. Let $f$ be the function such that $f\left(n\right)$ gives the $n$-th term of the sequence for positive integers $n\ge 1$.

The sequence values can also be expressed as the values of a linear function $L$ at consecutive integer inputs, so that $L\left(n\right)=f\left(n\right)$ for all positive integers $n$.

Construct an expression for $L\left(n\right)$. Show the work that leads to your answer.

In an arithmetic sequence $a_n$, the second term is $a_2=13$ and the fifth term is $a_5=34$.

Determine the common difference $d$ and the value of $a_7$. Show the work that leads to your answers.

All terms of a geometric sequence $g_n$ are positive. The first term is $g_1=5$ and the third term is $g_3=45$.

Determine the common ratio $r$ and the value of $g_5$. Show the work that leads to your answers.

The table gives the mass of a bacterial culture, in grams, at selected times $t$ hours after it was placed in a petri dish.

Determine whether the data is best modeled by a linear function or an exponential function. Give a reason for your answer. Then construct an appropriate function model $M\left(t\right)$.

The population of a city, in thousands of people, $t$ years after 2015 is modeled by the function $P\left(t\right)=120·e^{kt}$, where $k$ is a constant. In 2020 ($t=5$), the population was $165$ thousand.

Find the value of $k$ as a decimal approximation. Then use the model to find the predicted population of the city, in thousands, in 2030 ($t=15$). Express your answer as a decimal approximation.

The exponential function $f$ is given 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 $\left(0,7\right)$ and $\left(2,63\right)$.

Find the exact values of $a$ and $b$. Show the work that leads to your answers.

The function $h$ is given by $h\left(x\right)=14·{12}^{\frac{x}{3}}$.

Rewrite $h\left(x\right)$ in the equivalent form $h\left(x\right)=a·b^x$. Identify the exact value of $a$ and the value of $b$ as a decimal approximation.

A scientist measures a bacterial population at the start of each hour for three hours. The counts form a geometric sequence, as shown in the table.

The value $g_n$ gives the bacterial count at the start of hour $n$.

Construct an expression for $g_n$ for non-negative integers $n$. Show the work that leads to your answer.

The bacterial count varies continuously and can also be modeled by an exponential function $h$ of the form $h\left(t\right)=a·b^t$, where $t$ is the time in hours since the start of measurement ($t\ge 0$), and $a$ and $b$ are positive constants.

Construct the function $h\left(t\right)$ such that $h\left(n\right)=g_n$ for all non-negative integers $n$.

Use $h$ to find the value of $t$, as a decimal approximation, for which $h\left(t\right)=5000$.

The number of subscribers, in thousands, to a streaming service is recorded each year. The data for three selected years since the launch of the service is given in the table.

The number of subscribers can be modeled by an exponential function $S$ given by $S\left(t\right)=a·b^t$, where $S\left(t\right)$ is the number of subscribers, in thousands, $t$ is the number of years since the launch, and $a$ and $b$ are positive constants.

(i) Use the given data to write two equations that can be used to find the values for the constants $a$ and $b$ in the expression for $S\left(t\right)$.

(ii) Find the values for $a$ and $b$ as decimal approximations.

The function $f$ is given by $f\left(x\right)=\frac{4^{x+1}·8^x}{2^{2x-3}}$.

Rewrite $f\left(x\right)$ as an exponential expression with a base of $2$. Your result should be of the form $2^{mx+c}$, where $m$ and $c$ are integer constants. Identify the values of $m$ and $c$.