Arithmetic Series

Pairing Terms to Add the Numbers from One to One Hundred

A well-known anecdote about Carl Friedrich Gauss illustrates a clever way to add many numbers. In the story, a teacher asks the students to add all the numbers from $1$ to $100$: $1 + 2 + 3 + \dots + 98 + 99 + 100$.

Gauss did not add the terms one by one. He paired the beginning of the list with the end. A sum formed from the terms of an arithmetic sequence is called an arithmetic series.

For example, $1, 2, 3, \dots, 100$ is an arithmetic sequence with the first term $a=1$ and a common difference $b=1$. The corresponding arithmetic series is $1 + 2 + 3 + \dots + 100$.

Gauss Pairing Method

Gauss paired terms from opposite ends of the list:

• If the first term $(1)$ is added to the last term $(100)$, the result is $101$.
• If the second term $(2)$ is added to the second-to-last term $(99)$, the result is also $101$.
• If the third term $(3)$ is added to the third-to-last term $(98)$, the result is still $101$.
• The same pattern continues to the middle of the list.

There are $50$ pairs, and every pair sums to $101$. Therefore, the total is $50 \times 101 = 5050$.

Finding the General Formula

We can use Gauss's method to derive a general formula for the sum of the first $n$ terms of an arithmetic series, usually denoted by $S_n$.

Begin with an arithmetic series:

If written out with the first term $(a)$ and the common difference $(b)$:

Write the same series $S_n$ in reverse order, from the last term to the first:

Or:

Add the two representations of $S_n$ term by term:

Each vertical pair has the same sum, $2a + (n-1)b$. Since each row contains $n$ terms, the addition produces $n$ identical pair sums.

So, we get:

By dividing both sides by $2$, we obtain the formula for the sum of the first $n$ terms of an arithmetic series:

The derivation works for both even and odd $n$. When $n$ is odd, reversing and adding the series counts the middle term correctly because both copies of the series are included.

Practical Formulas for Arithmetic Series

There are two main formulas commonly used to calculate $S_n$:

1. If the first term $(a)$ and the common difference $(b)$ are known:

   $$S_n = \frac{n}{2}(2a + (n - 1)b)$$

2. If the first term $(a)$ and the $(n)$-th term $(U_n)$ are known: Recall the formula for the $n$-th term is . Substituting this into the first formula:

Notation:

• $S_n$ = Sum of the first $n$ terms
• $n$ = Number of terms
• $a$ = First term ($U_1$)
• $b$ = Common difference (difference between terms)
• $U_n$ = Term number $n$

Calculating Two Arithmetic Series

First Problem

Recalculate the sum of the series $1 + 2 + 3 + \dots + 98 + 99 + 100$.

Given:

• First term $a = 1$
• Last term $U_n = U_{100} = 100$
• Number of terms $n = 100$

Since $a$ and $U_n$ are known, we use the second formula:

This agrees with the earlier pairing calculation.

Second Problem

Given the arithmetic series: $13 + 16 + 19 + 22 + \dots$. Calculate the sum of the first $30$ terms $(S_{30})$!

Given:

• First term $a = 13$
• Common difference $b = 16 - 13 = 3$
• Number of terms to sum $n = 30$

Since $a$ and $b$ are known, we use the first formula:

So, the sum of the first $30$ terms of this series is $1695$.

Application to Theater Seating

A theater has $20$ seats in the first row and $4$ more seats in each following row. The number of seats in row $15$ and the total in the first fifteen rows are:

The first result describes only row $15$. The second result describes all seats across the first fifteen rows, so the sequence term and the series sum answer different questions.

Exercise

Calculate the finite arithmetic series:

Solution

First determine how many terms the series contains. The common difference is $5$:

The sum is $1110$.