A geometric series is the sum of the terms of a geometric sequence. A geometric sequence is a sequence where each term is found by multiplying the previous term by a constant called the common ratio ($r$). A finite geometric series has a specific number of terms. The sum of a finite geometric series can be found using a formula that depends on the first term ($a$), the common ratio ($r$), and the number of terms ($n$). This worksheet will guide you through deriving and applying this formula.
The formula for the sum ($S_n$) of a finite geometric series is given by:
$S_n = a \frac{1 - r^n}{1 - r}$, where $r \neq 1$
Match the following terms with their definitions:
| Term | Definition |
|---|---|
| 1. Common Ratio | A. The sum of the terms in a geometric sequence up to a certain point. |
| 2. Geometric Sequence | B. A sequence where each term is multiplied by a constant to get the next term. |
| 3. Finite Geometric Series | C. The value by which each term is multiplied to get the next term in a geometric sequence. |
| 4. First Term | D. The initial value in a geometric sequence. |
| 5. Sum of a Series | E. A geometric series with a limited number of terms. |
Complete the following paragraph using the words provided (first term, common ratio, sum, terms, finite).
A _______ geometric series is the _______ of a geometric sequence with a limited number of _______. To find the _______ of this series, we need to know the _______ and the _______.
Explain in your own words why the formula $S_n = a \frac{1 - r^n}{1 - r}$ cannot be used when $r = 1$. What happens to the series in this specific case, and how would you calculate the sum?
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