# Not Just Calculus: First-Year Introduction to Upper-Level Mathematics

## Section1.4Linear and Exponential Growth

Let $$b,m$$ be real constants, and consider the linear function $$L(t)=b+mt\text{.}$$ The sequence of values $$L(0), L(1), L(2),\ldots$$ given by
\begin{equation*} b, b+m, b+2m,\ldots ,b+nm,\ldots \end{equation*}
is called an arithmetic sequence
1
The emphasis is on the third syllable "met" when the word "arithmetic" is used as an adjective rather than a noun. For example: "Addition is an operation of $$\mbox{a}\cdot \mbox{rith}'\cdot \mbox{metic}\text{.}$$ Repeated addition creates an $$\mbox{arith}\cdot \mbox{met}'\cdot \mbox{ic}$$ sequence."
with initial term $$b$$ and common difference $$m\text{.}$$ An arithmetic sequence is said to exhibit linear growth or decay, according to whether $$m \gt 0$$ or $$m\lt 0\text{,}$$ respectively.
Let $$a,r$$ be real constants with $$a\neq 0, r\gt 0, r\neq 1\text{,}$$ and consider the exponential function $$E(t) = ar^t\text{.}$$ The sequence of values $$E(0),E(1),E(2),\ldots$$ given by
\begin{equation*} a,ar,ar^2,\ldots,ar^n,\ldots \end{equation*}
is called a geometric sequence with initial term $$a$$ and common ratio $$r\text{.}$$ A geometric sequence with $$a\gt 0$$ is said to exhibit exponential growth or decay, according to whether $$r\gt 1$$ or $$r\lt 1\text{,}$$ respectively.

### Checkpoint1.4.1.

Fill in the missing terms of the following arithmetic and geometric sequences. Identify the initial term and the common difference or common ratio for each.
1. $$\displaystyle 5,2,-1, \underline{\rule{2ex}{0ex}}, \underline{\rule{2ex}{0ex}},\underline{\rule{2ex}{0ex}},\ldots$$
2. $$\displaystyle 5,2,0.8,\underline{\rule{2ex}{0ex}}, \underline{\rule{2ex}{0ex}},\underline{\rule{2ex}{0ex}},\ldots$$
3. $$\displaystyle \underline{\rule{2ex}{0ex}}, 2, \underline{\rule{2ex}{0ex}},5,\underline{\rule{2ex}{0ex}},8,\ldots$$
4. $$\displaystyle \underline{\rule{2ex}{0ex}}, 2, \underline{\rule{2ex}{0ex}},4,\underline{\rule{2ex}{0ex}},8,\ldots$$
Finite arithmetic and geometric sums. Exercises at the end of this subsection outline the derivations of the following formulas.
\begin{align} b + (b+m) + (b+2m) + \cdots + (b+nm) \amp = \frac{(n+1)(2b+nm)}{2} \tag{1.4.1}\\ a + ar + ar^2 + \cdots + ar^n \amp = a\left(\frac{1-r^{n+1}}{1-r}\right)\tag{1.4.2} \end{align}

### Checkpoint1.4.2.

Find the given sums of terms of arithmetic and geometric sequences.
1. Find the sum of the first 100 positive integers.
2. $$\displaystyle 2 + 5 + 8 + 11 + \cdots + 302$$
3. $$\displaystyle 2 + 5 + 8 + 11 + \cdots + 1571$$
4. $$\displaystyle 2 + 6 + 18 + 54 + \cdots + 2(3^{100})$$
5. $$\displaystyle 2 + 6 + 18 + 54 + \cdots + 9565938$$
Infinite geometric sums. An infinite sum of the form
\begin{equation*} a + ar + ar^2 + ar^3 + \cdots \end{equation*}
is called an infinite geometric series, and is defined to mean $$\displaystyle \lim_{n\to\infty} s_n$$ (if the limit exists), where $$s_1,s_2,s_3,\ldots$$ is sequence of finite sums
\begin{align*} s_0 \amp = a\\ s_1 \amp = a + ar\\ s_2 \amp = a + ar + ar^2\\ \amp \vdots \\ s_n \amp = a+ ar+ar^2 + \cdots + ar^n\\ \amp \vdots \end{align*}
If $$|r| \lt 1\text{,}$$ then $$|r|^n \to 0$$ as $$n\to \infty\text{.}$$ Using properties of limits from calculus, we have
\begin{equation*} a\left(\frac{1-r^{n+1}}{1-r} \right) \to a\left(\frac{1}{1-r}\right) \end{equation*}
as $$n\to \infty\text{.}$$ Putting this together with (1.4.2) above is the justification for the following formula.
\begin{align} a + ar+ ar^2 + ar^3 + \cdots = a\left(\frac{1}{1-r}\right) \amp \amp \mbox{ for } |r|\lt 1\tag{1.4.3} \end{align}

### ExercisesExercises

#### 1.

Derive (1.4.1).
Hint.
Write the sum in reverse order $$L(n)+L(n-1) + \cdots + L(1) + L(0)$$ directly beneath $$L(0)+ L(1) + \cdots + L(n)\text{,}$$ in such a way that the terms are aligned vertically. Notice that each vertically aligned pair has the form $$L(k)$$ and $$L(n-k)\text{,}$$ and that $$L(k)+L(n-k) = 2b+nm$$ (the $$k$$’s cancel!). Now go from there.

#### 2.

Derive (1.4.2).
Hint.
Let $$s$$ be the desired sum $$a+ar+ar^2 + \cdots + ar^n\text{.}$$ Examine the expansion of $$s-rs$$ (many terms cancel!). Simplify and solve for $$s\text{.}$$