When you're learning about sequences and series in Grade 12 Pre-Calculus, it’s super important to know the difference between geometric series and arithmetic series. At first, they might look pretty similar, but they actually behave very differently in how their terms are created, added up, and whether they get bigger forever or settle down to a number.

Definitions

1. Arithmetic Series: An arithmetic series is made by adding numbers that have the same difference between them. For example, in the series $2, 5, 8, 11$, we keep adding $3$ each time. So, we can write it like this:

   $S_n = a + (a + d) + (a + 2d) + \ldots + (a + (n-1)d)$

   Here, $a$ is the first number in the series, $d$ is the fixed amount we add, and $n$ is how many numbers are in the series.

2. Geometric Series: A geometric series, on the other hand, is created by multiplying each number by the same factor. For example, in the series $3, 6, 12, 24$, each number is made by multiplying the previous one by $2$. We can write a geometric series like this:

   $S_n = a + ar + ar^2 + ar^3 + \ldots + ar^{n-1}$

   Here, $a$ is the first number, $r$ is the factor we multiply by, and $n$ is how many numbers are in the series.

Behavior Comparison

Growth Rate

• Arithmetic Series: The growth of an arithmetic series is steady and simple. For instance, in the series $2, 5, 8, 11$, we add $3$ each time, so the sum grows slowly, adding the same amount each time.

  • Formula: You can find the sum of the first $n$ terms using this formula:

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

    For our example, if we add up the first $10$ terms of $2, 5, 8, 11,\ldots$, we can calculate it like this:

    $S_{10} = \frac{10}{2} \times (2 \times 2 + (10 - 1) \times 3) = 5 \times (4 + 27) = 155$

• Geometric Series: Now, a geometric series grows a lot faster if the factor $r > 1$. For the series $3, 6, 12, 24$, the total can grow really fast as we add more terms.

  • Formula: You can find the sum of the first $n$ terms with this formula:

    $S_n = a \frac{1-r^n}{1-r}$ for $r \neq 1$

    Using the series $3, 6, 12, 24$, if we look at the first $4$ terms, it would be:

    $S_4 = 3 \frac{1-2^4}{1-2} = 3 \times \frac{1 - 16}{-1} = 3 \times 15 = 45$

Finite vs. Infinite Series

• Finite Arithmetic Series: The sum of a finite arithmetic series is easy to find, as we showed earlier.

• Infinite Geometric Series: An infinite geometric series behaves differently. If $|r| < 1$, the infinite sum can settle down to a fixed number given by:

  $S = \frac{a}{1 - r}$

  For example, consider the series $1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \ldots$, where $a = 1$ and $r = \frac{1}{2}$. The total would be:

  $S = \frac{1}{1 - \frac{1}{2}} = 2$

  But, if $|r| \geq 1$, the series will not settle down and just keep getting bigger, which is different from arithmetic series since they will always grow indefinitely when they have an infinite number of terms.

Conclusion

In summary, arithmetic series grow steadily and are easy to sum up. Meanwhile, geometric series grow quickly and have a cool concept called convergence when we look at them infinitely. Learning the differences between these types of series not only helps you understand math better, but also gets you ready for more advanced topics in calculus and beyond!