Lesson 5.2: Arithmetic Series
Introduction
In this lesson, we will delve into the concept of arithmetic series, a fundamental topic in mathematics that has practical applications in finance, economics, and everyday life. students, by the end of this lesson, you will understand:
- The nth term of an arithmetic sequence.
- The sum of an arithmetic series.
- The applications of arithmetic series to regular, fixed increases.
To hook your interest, consider this: Whether you’re saving for a new bike or planning your monthly budget, understanding how different amounts accumulate over time can help you make informed financial decisions. Let's get started!
The nth Term of an Arithmetic Sequence
An arithmetic sequence is a list of numbers in which the difference between consecutive terms is constant. This difference is known as the common difference, denoted as $d$.
Definition
The general formula to find the nth term $a_n$ of an arithmetic sequence is given by:
$$
a_n = a_1 + (n - 1)d
$$
where:
- $a_n$ is the nth term,
- $a_1$ is the first term,
- $d$ is the common difference,
- $n$ is the term number.
Example 1
Let's consider an arithmetic sequence where the first term $a_1 = 3$ and the common difference $d = 2$. To find the 5th term $a_5$, we can apply the formula:
$$
a_5 = 3 + (5 - 1) $\cdot 2$
$$
Calculating this gives:
$$
a_5 = 3 + $4 \cdot 2$ = 3 + 8 = 11
$$
Hence, the 5th term of this series is 11.
The Sum of an Arithmetic Series
An arithmetic series is the sum of the terms of an arithmetic sequence. The formula to calculate the sum $S_n$ of the first $n$ terms of an arithmetic series is:
$$
$ S_n = \frac{n}{2} (a_1 + a_n)$
$$
This can also be expressed as:
$$
S_n = $\frac{n}{2}$ [2a_1 + (n - 1)d]
$$
Example 2
Continuing with the sequence where $a_1 = 3$ and $d = 2$, let's calculate the sum of the first 5 terms. First, we need to find $a_5$ again or use our previous result directly:
$$
$ S_5 = \frac{5}{2} (3 + 11)$
$$
This simplifies to:
$$
S_5 = $\frac{5}{2}$ $\cdot 14$ = $\frac{70}{2}$ = 35
$$
Thus, the sum of the first 5 terms of the series is 35.
Applications of Arithmetic Series to Regular, Fixed Increases
Arithmetic series provide essential insights in real-life situations, especially in financial mathematics. Regularly deposited amounts, varying salaries, or even savings grow in arithmetic manner under certain conditions.
Example 3: Saving for a Goal
Suppose you want to save for a laptop costing $1,200. You decide to save $100 each month, starting in January. Here’s how we can model the situation with an arithmetic series:
- The first month saves $100: $a_1 = 100$
- Each subsequent month you add another $100, so $d = 100$
- Let’s find how long it will take to save $1,200.
First, calculate the number of terms $n$ until the total savings reach $1,200.
Using the sum formula:
$$
S_n = $\frac{n}{2}$ [$2 \cdot 100$ + (n - 1) $\cdot 100$] = 1200
$$
This expands to:
$$
$\frac{n}{2}$ [200 + (n - 1) $\cdot 100$] = 1200
$$
Multiplying through by 2 gives:
$$
n [200 + (n - 1) $\cdot 100$] = 2400
$$
Distributing, we get:
$$
200n + 100n^2 - 100n = 2400
$$
Combining like terms leads to:
$$
100n^2 + 100n - 2400 = 0
$$
Now we can simplify by dividing the entire equation by 100:
$$
n^2 + n - 24 = 0
$$
To solve this quadratic equation, we use the quadratic formula:
$$
n = $\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
$$
In our case, $a = 1$, $b = 1$, and $c = -24$. Plugging in these values gives:
$$
n = $\frac{-1 \pm \sqrt{1 + 96}}{2}$ = $\frac{-1 \pm 10}{2}$
$$
Calculating the two potential roots:
- $ n = \frac{9}{2} = 4.5 \quad \text{(not possible as n must be whole)}$
- $ n = \frac{-11}{2} \quad \text{(not applicable)}$
The nearest whole number is $n = 5$, suggesting you save for 5 months.
Verifying through substitution, $S_5 = 1200$. Thus, within 5 months, you'll save enough for your laptop.
Conclusion
Arithmetic series not only help us understand the behavior of numbers but also apply directly to real-world financial scenarios. By mastering the nth term and the sum of an arithmetic series, you will be well-equipped to tackle issues related to savings, loans, and investments.
Study Notes
- An arithmetic sequence has a constant difference between terms, denoted as $d$.
- The nth term is calculated using $a_n = a_1 + (n - 1)d$.
- The sum of an arithmetic series can be computed with $S_n = \frac{n}{2} (a_1 + a_n)$ or $S_n = \frac{n}{2} [2a_1 + (n - 1)d]$.
- Arithmetic series are crucial in financial mathematics, illustrating concepts such as regular savings and loans.