Problem 31 - College Algebra

Question

Find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence. $$ \sum_{i=1}^{8} 3^{i} $$

Step-by-Step Solution

Answer

The sum of the first 8 terms of the given geometric sequence is 9,842

Step 1: Determine the first term \( a_1 \)

In this particular sequence, we can see that the first term \( a_1 \) is \( 3^1 = 3 \)

Step 2: Identify the common ratio r

The common ratio r is simply 3, based on the given sequence.

Step 3: Identify the number of terms n

The number of terms n in this sequence is 8, as given by the sum notation.

Step 4: Apply the formula for the sum of a geometric sequence

Now, we can use the formula \( S_n = a_1 \cdot \frac{1 - r^n}{1-r} \) to calculate the sum. Substituting the values we have: \( S_8 = 3 \cdot \frac{1 - 3^8}{1-3} \)

Step 5: Calculate the sum

Perform the calculations: \( S_8 = 3 \cdot \frac{1 - 6,561}{-2} = 9,842 \)

Key Concepts

Sum of a Geometric Series

When we talk about the sum of a geometric series, we refer to the total of all the terms in a sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. The formula to find the sum of the first n terms, often represented as \( S_n \), is particularly simple yet powerful:

\[ S_n = a_1 \cdot \frac{1 - r^n}{1-r} \]
where \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.

• \( a_1 \) - We start by identifying the first term in the series. In our exercise, \( a_1 = 3 \).
• \( r \) - Next, determine the common ratio; for the given series \( r = 3 \).
• \( n \) - Count the number of terms, which is 8 in this context.

Once we have these, we apply the values to our formula and calculate, leading us to the sum of the series. It's essential to pay attention to the signs when plugging in the values to avoid common mistakes like miscalculating when the ratio is negative or greater than 1, which significantly alters the sum.

Geometric Progression

A geometric progression, also known as a geometric sequence, is a series of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio. This type of sequence occurs frequently in real-world situations, like in the growth of populations, compound interest calculations, and fractal patterns.

Here is the sequence structure in general form:
\[ a, a \cdot r, a \cdot r^2, a \cdot r^3, ... \]
Each term can be found using the formula \( a_n = a_1 \cdot r^{(n-1)} \), where \( a_n \) is the nth term. For example, in the given exercise, the sequence we have is:

• \( 3, 3 \cdot 3^1, 3 \cdot 3^2, ..., 3 \cdot 3^7 \) for the first 8 terms.

The geometric progression concept is pivotal to understanding growth patterns and helps in predicting future behavior when the terms are related to time or other sequential factors. In finance, geometry progressions are the core behind understanding loans, investments, and annuities.

Sequence and Series Algebra

The studies of sequences and series form an essential part of algebra, particularly when dealing with ordered sets of numbers and their sums. They are not only critical for mathematical analysis but also have practical applications in various disciplines.

A sequence is a set of numbers in a specific order, while a series is the sum of the elements of a sequence. Two main types of series are arithmetic and geometric series, with each having a unique set of properties and formulae for analysis.

• In an arithmetic series, the difference between consecutive terms is constant.
• In a geometric series, the ratio between consecutive terms is constant.

Understanding the algebra of sequences and series enables us to solve complex problems by breaking them down into simpler, more manageable sequences. When confronted with a series like \( \sum_{i=1}^{8} 3^{i} \), students should remember to carefully determine the first term, common ratio (or difference in the case of arithmetic), and apply the correct formula to find sums or individual terms. It's worth taking the time to thoroughly understand these concepts, as they form the foundation for higher levels of mathematics and its applications in the real world.