Difference between Arithmetic and Geometric Series

Arithmetic Series

Basic concept:

An arithmetic series is the sum of the terms of an arithmetic sequence. Remember, an arithmetic sequence is one that has a constant difference (common difference) between its terms ( $b$ ).

So, we are summing terms with the pattern: $a, a+b, a+2b, a+3b, \dots$ .

The sum of the first $n$ terms ( $S_n$ ) of an arithmetic series can be calculated using the formula:

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

An equivalent formula is:

S_n = \frac{n}{2}(a + U_n)

Where $a$ is the first term and $U_n$ is the $n$ -th term.

Imagine you are stacking bricks. The first layer has $1$ brick, the second layer has $3$ bricks, the third layer has $5$ bricks, and so on (common difference $= 2$ ). An arithmetic series represents the total number of bricks needed to make a stack $n$ layers high.

Geometric Series

Basic concept:

A geometric series is the sum of the terms of a geometric sequence. Remember, a geometric sequence is one that has a constant ratio (common ratio) between its terms ( $r$ ).

So, we are summing terms with the pattern: $a, ar, ar^2, ar^3, \dots$ .

The sum of the first $n$ terms ( $S_n$ ) of a geometric series can be calculated using the formula:

S_n = \frac{a(r^n - 1)}{r-1}

This formula applies when $r \neq 1$ . In the formula, $a$ is the first term and $r$ is the common ratio.

If $r=1$ , every term equals $a$ , so the special-case sum is:

S_n=na

Suppose a workshop produces $1$ , then $2$ , then $4$ new components in successive hours. The hourly output has ratio $r=2$ . A geometric series gives the cumulative number of newly produced components over $n \text{ hours}$ . During the first , the workshop produces in total.

Arithmetic and Geometric Series

Feature	Arithmetic Series	Geometric Series
Basis	Sum of terms in an arithmetic sequence (common difference $b$ )	Sum of terms in a geometric sequence (common ratio $r$ )

Calculating Arithmetic and Geometric Sums

For the arithmetic series:

5+8+11+14+17

This series has the common difference $b=3$ . Its first five terms sum to:

S_5=\frac52(5+17)=55

For the geometric series:

5+10+20+40+80

This series has the common ratio $r=2$ . Its first five terms sum to:

S_5=5\frac{2^5-1}{2-1}=155

Patterns of Terms and Partial Sums

The terms of an arithmetic series change linearly with the index, while its partial sums generally change quadratically. The terms of a geometric series change exponentially, while its finite partial sums follow the geometric sum formula.

The name of a series comes from its sequence of terms. The partial sums $S_1,S_2,S_3,\ldots$ can follow a different pattern.

Exercise

Classify each series and calculate its sum:

$30+25+20+15+10$
$2+6+18+54+162$

Solution

The first series is arithmetic with $b=-5$ . The second is geometric with $r=3$ :

S_5=\frac52(30+10)=100