How to find the sum of a geometric sequence?
Understand the Problem
The question is asking how to calculate the sum of a geometric sequence, which involves understanding the formula for the sum of such sequences and the parameters involved, such as the first term, the common ratio, and the number of terms.
Answer
The formula for the sum of a geometric sequence is $S_n = a \frac{1 - r^n}{1 - r}$.
Answer for screen readers
The sum of the geometric sequence is given by $S_n = a \frac{1 - r^n}{1 - r}$.
Steps to Solve
- Identify the parameters of the geometric sequence
To calculate the sum of a geometric sequence, you need to know the first term ($a$), the common ratio ($r$), and the number of terms ($n$).
- Use the sum formula for a geometric series
The formula for the sum of the first $n$ terms of a geometric sequence is given by:
$$ S_n = a \frac{1 - r^n}{1 - r} $$
This formula is used when $r \neq 1$. If $r = 1$, then the sum becomes $S_n = n \cdot a$.
- Plug in the values into the formula
Substituting the values of $a$, $r$, and $n$ into the formula will give you the sum of the geometric sequence.
- Simplify the expression
After substituting the values, carry out the calculation by following the order of operations (parentheses, exponents, multiplication and division, addition and subtraction).
The sum of the geometric sequence is given by $S_n = a \frac{1 - r^n}{1 - r}$.
More Information
The sum of a geometric sequence is a useful calculation in many real-life scenarios, such as calculating compound interest in finance or predicting population growth.
Tips
- Forgetting to check if $r = 1$ before applying the formula.
- Not correctly substituting values into the sum formula.
- Misunderstanding the common ratio and first term.