Problem 53 - Algebra 2

Question

What is the common ratio for the geometric series \(\sum_{n=1}^{10} 7\left(\frac{4}{7}\right)^{n-1} ?\) Enter your answer as a fraction.

Step-by-Step Solution

Answer

The common ratio of the geometric series is \(\frac{4}{7}\).

Step 1: Identify the geometric series

The geometric series in this exercise is \(\sum_{n=1}^{10} 7\left(\frac{4}{7}\right)^{n-1}\).

Step 2: Determine the common ratio

The common ratio (r) is the value being raised to the power of \(n-1\). Looking at the geometric series, the common ratio is \(\frac{4}{7}\).

Key Concepts

Common Ratio

In a geometric series, the common ratio is a key element that determines how the terms progress. It is the factor by which each term is multiplied to get the next term. Identifying the common ratio helps us understand the pattern of the series.
The common ratio is often represented by the letter \( r \) and is constant throughout the series.

• In the given series \( \sum_{n=1}^{10} 7\left(\frac{4}{7}\right)^{n-1} \), look closely at the expression inside the series.
• The value that is raised to the power of \( n-1 \) is the common ratio.

For this specific series, the common ratio is shown as \( \frac{4}{7} \).
A good tip is to always look for this consistent multiplier when identifying geometric series, as it reveals the steady growth or decay of the series.

Geometric Sequence

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. This sequence can appear in various forms, including finite and infinite sequences.
In the context of a geometric series, which is the sum of the terms of a geometric sequence, understanding the sequence helps us predict and find values. Here's how it works:

• The first term is multiplied by the common ratio to get the second term.
• The second term is multiplied by the common ratio again to get the third term, and so on.

Given the sequence from the exercise,\( 7, 7 \times \frac{4}{7}, 7 \times \left(\frac{4}{7}\right)^2 \), and so forth, each term follows this specific multiplication pattern.
Understanding this pattern is crucial in solving problems involving geometric sequences.

Algebra 2

Algebra 2 is a branch of mathematics that builds upon the concepts learned in Algebra 1, introducing students to more complex functions and advanced topics such as geometric sequences and series.
In Algebra 2, geometric sequences are often explored deeper as students learn to solve problems involving formula and sequence patterns. This involves:

• Understanding how geometric sequences form the basis for series.
• Applying formulas to find specific terms within sequences.
• Having the ability to calculate the sum of a series, both finite and infinite.

By mastering these skills, students can solve real-world problems where patterns of growth or decay appear, such as in financial calculations or scientific scenarios.
In the given exercise, identifying the common ratio and recognizing the structure of the series are part of the broader learning objectives in Algebra 2.