Problem 60 - Algebra 2

Question

REASONING Explain how to write the series \(2+12+72+432+2592\) using sigma notation.

Step-by-Step Solution

Answer

\(\sum_{n=1}^{5} 2 \cdot 6^{(n-1)}\)

Step 1: Identify the Pattern in the Series

Firstly, identify how each term is related to the next in the series: \(2, 12, 72, 432, 2592\). Start with 2 and see how it changes: \(2 \times 6 = 12\), \(12 \times 6 = 72\), \(72 \times 6 = 432\), \(432 \times 6 = 2592\). Each term is obtained by multiplying the previous term by 6.

Step 2: Determine the Starting Point and General Formula

We see that the first term is 2. Therefore, the series can be defined using a geometric sequence where the first term \(a = 2\) and the common ratio \(r = 6\). The general term of a geometric sequence is \(a_n = a \cdot r^{(n - 1)}\). Here \(a = 2\) and \(r = 6\), so \(a_n = 2 \cdot 6^{(n - 1)}\).

Step 3: Determine the Number of Terms in the Series

We need to find how many terms compose the series. Count the terms: 2, 12, 72, 432, 2592. There are 5 terms in this series.

Step 4: Write the Series in Sigma Notation

Use the sigma notation to sum the terms: \(\sum_{n=1}^{5} 2 \cdot 6^{(n-1)}\). This expression sums all terms from \(n=1\) to \(n=5\) using the general term formula we found.

Key Concepts

Geometric Series

A geometric series is a way of expressing a sum of numbers where each term grows by a specific factor. This factor is consistent across the series. Consider the series given: 2, 12, 72, 432, 2592. Each of these numbers can be seen as part of a larger pattern where each number is obtained by multiplying the previous one by 6.
When writing a geometric series in sigma notation, you are essentially expressing the sum of these terms. In this example, you can represent our series as \( \sum_{n=1}^{5} a_n \), where each \( a_n \) is given by a specific formula based on its position in the series. The formula is typically in the form of \( a \cdot r^{(n-1)} \), where \( a \) is the first term and \( r \) is the common ratio.

• In our series, the sum becomes \( 2 + 12 + 72 + 432 + 2592 \), neatly expressed in sigma notation as \( \sum_{n=1}^{5} 2 \cdot 6^{(n-1)} \).

Understanding the structure of a geometric series helps in effectively utilizing sigma notation to simplify expressions.

Common Ratio

The common ratio in a geometric sequence or series is the factor by which each term is multiplied to get the next term. For the series 2, 12, 72, 432, 2592, you notice that you can multiply 2 by 6 to get 12, multiply 12 by 6 to get 72, and so on. Therefore, the common ratio \( r \) is 6.
The common ratio plays a crucial role because it determines the exponential growth rate of the series. Knowing the common ratio, you can generate any term in the series without listing all previous terms.

• The sequence formed by a common ratio can be expressed as \( a, ar, ar^2, ar^3, \ldots \)
• In our example, the progression is \( 2, 12, 72, 432, 2592 \), with \( r = 6 \).

Calculating the common ratio is an essential first step in identifying the properties of the geometric series.

General Term

The general term of a geometric sequence helps to describe any term in the sequence using its position. This is very useful for identifying patterns and simplifying calculations.
The formula for the general term of a geometric sequence is \( a_n = a \cdot r^{(n-1)} \). Here, \( a \) is the first term of the sequence, \( r \) is the common ratio, and \( n \) is the term number.

• In our series \( a = 2 \) and \( r = 6 \).
• This gives us \( a_n = 2 \cdot 6^{(n-1)} \)

Using this formula, we can calculate any term in the sequence without calculating all the preceding terms. It makes understanding and working with sequences much easier.

Geometric Sequence

A geometric sequence is a list of numbers where each number is derived by multiplying the previous one by a constant called the common ratio. This type of sequence shows up in many contexts, from finance to science.
In this sequence, the numbers 2, 12, 72, 432, and 2592 are obtained by consistently multiplying by the common ratio of 6.

• In a geometric sequence, the pattern of growth determines future terms.
• It is defined by its first term and the common ratio.

Just knowing these two pieces of information lets you construct the entire sequence or any part of it. Understanding geometric sequences helps in recognizing such patterns in various mathematical and real-world applications.