Understanding Arithmetic Series

An arithmetic series is simply the sum of the terms of an arithmetic sequence. While a sequence is a list of numbers with a common difference (like $2, 4, 6, 8$), a series adds those numbers together (like $2 + 4 + 6 + 8$).

The Arithmetic Series Formulas

There are two main formulas to find the sum of the first $n$ terms of an arithmetic series, denoted as $S_n$.

Formula 1: When you know the first and last terms $S_n = \frac{n}{2}(a_1 + a_n)$

Formula 2: When you know the first term and the common difference $S_n = \frac{n}{2}[2a_1 + (n-1)d]$

Where:

• $S_n$ is the sum of the first $n$ terms.
• $n$ is the number of terms.
• $a_1$ is the first term.
• $a_n$ is the $n$-th (or last) term.
• $d$ is the common difference between consecutive terms.

Example 1: Sum of the First 50 Positive Integers

Problem: Find the sum of the first 50 positive integers ($1 + 2 + 3 + \dots + 50$).

Solution:

1. Identify the known values:

   • Number of terms, $n = 50$
   • First term, $a_1 = 1$
   • Last term, $a_{50} = 50$

2. Since we know the first and last terms, we can use Formula 1: $S_{50} = \frac{50}{2}(1 + 50)$ $S_{50} = 25(51)$ $S_{50} = 1275$

The sum of the first 50 positive integers is $1275$.

Example 2: Using Sigma Notation

Problem: Find the sum of the series written in sigma notation: $\sum_{k=1}^{20} (3k + 1)$

Solution: This notation means we are adding the terms generated by the formula $3k + 1$ from $k = 1$ up to $k = 20$.

1. Find the number of terms ($n$): From $k=1$ to $20$, there are $n = 20$ terms.

2. Find the first term ($a_1$) by plugging in $k=1$: $a_1 = 3(1) + 1 = 4$

3. Find the last term ($a_{20}$) by plugging in $k=20$: $a_{20} = 3(20) + 1 = 61$

4. Use Formula 1: $S_{20} = \frac{20}{2}(4 + 61)$ $S_{20} = 10(65)$ $S_{20} = 650$

The sum of the series is $650$.