91Ó°ÊÓ

When trying to find the sum of a geometric series, it helps to understand the structure of the series itself. A geometric series is a series of terms that have a constant ratio between consecutive terms.
In this case, we are working with the series \( \sum_{k=1}^{90} \frac{5}{7^{k}} \). Finding the sum of this series essentially means adding up all the terms from when \( k = 1 \) to \( k = 90 \).
The formula for the sum of the first \( n \) terms of a geometric series is given by:

• \[ S = \frac{a(1 - r^n)}{1 - r} \]

where \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms. This formula helps to calculate the total sum quickly without calculating the sum manually term by term.

The "common ratio" is key to understanding a geometric series. In a geometric series, the common ratio is the factor by which we multiply each term to get to the next term.
For example, in the series \( \sum_{k=1}^{90} \frac{5}{7^{k}} \), the common ratio \( r \) is \( \frac{1}{7} \).
This means that each term is \( \frac{1}{7} \) of the previous term. Knowing the common ratio is crucial because it allows us to use the formula for the sum of the series.
Quick Tips:

• Identify the pattern of multiplication in the sequence.
• Divide a subsequent term by its preceding term to find the common ratio.

Understanding this concept helps us manage and evaluate geometric progressions efficiently.

To evaluate any sequence, identifying the first term is crucial. The first term, often denoted as \( a \), is the starting point from which all other terms are derived.
In our series \( \sum_{k=1}^{90} \frac{5}{7^{k}} \), the first term is found by substituting \( k = 1 \) into the general term.
So, \( a = \frac{5}{7^1} = \frac{5}{7} \). This first term serves as the foundation for calculating the series sum using the formula.
Finding \( a \) is usually straightforward:

• Locate the first term in a series, which means the smallest index (here, \( k = 1 \)).
• Evaluate the expression for this first term.

Knowing the first term helps us apply the sum formula accurately and understand the sequence's scale.

Understanding the series evaluation method is crucial for solving geometric series problems effectively. When you evaluate a geometric series, like \( \sum_{k=1}^{90} \frac{5}{7^{k}} \), here's a practical approach to follow. First, identify the common ratio \( r \) and the first term \( a \) as fundamental components.
In our series, \( r = \frac{1}{7} \) and \( a = \frac{5}{7} \).
Next, determine the number of terms, \( n \), which is the count of terms you are summing. Here, the series runs from \( k = 1 \) to \( k = 90 \), giving us \( n = 90 \). Once you have \( a \), \( r \), and \( n \), plug them into the sum formula:

• \[ S = \frac{a(1 - r^n)}{1 - r} \]

This formula directly computes the sum by compensating for the series' growth determined by \( r \) and starting at \( a \). Simplify this expression by performing any cancellations or arithmetic transformations needed to reach a final straightforward result. By following these steps, you evaluate the sum efficiently, ensuring an accurate solution.