Problem 50 - Precalculus: Mathematics for Calculus

Question

Find the partial sum \(S_{n}\) of the geometric sequence that satisfies the given conditions. $$a=\frac{2}{3}, \quad r=\frac{1}{3}, \quad n=4$$

Step-by-Step Solution

Answer

The partial sum \( S_4 \) is \( \frac{80}{81} \).

Step 1: Identify the formula for the partial sum of a geometric sequence

The partial sum of the first \( n \) terms of a geometric sequence is given by the formula: \[ S_n = a \frac{1 - r^n}{1 - r} \] where \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.

Step 2: Substitute values into the formula

Use the values \( a = \frac{2}{3} \), \( r = \frac{1}{3} \), and \( n = 4 \) and substitute them into the formula: \[ S_4 = \frac{2}{3} \frac{1 - \left( \frac{1}{3} \right)^4}{1 - \frac{1}{3}} \]

Step 3: Simplify the denominator

Calculate the denominator: \( 1 - \frac{1}{3} = \frac{2}{3} \).

Step 4: Simplify the numerator

Calculate the numerator: \[ 1 - \left( \frac{1}{3} \right)^4 = 1 - \frac{1}{81} = \frac{80}{81} \]

Step 5: Complete the division

Substitute the simplified numerator and denominator back into the equation: \[ S_4 = \frac{2}{3} \times \frac{80}{81} \div \frac{2}{3} = \frac{80}{81} \]

Step 6: Write the final answer

Thus, the partial sum \( S_4 \) of the sequence is \( \frac{80}{81} \).

Key Concepts

Partial Sum

In a geometric sequence, the partial sum is the total of the first few terms of the sequence. Specifically, it is the sum of the first \(n\) terms. Calculating the partial sum is useful when wanting to find the sum up to a certain point in the sequence without adding each term manually.

The formula to calculate the partial sum \(S_n\) for a geometric sequence is:

• \(S_n = a \frac{1 - r^n}{1 - r}\)

Here, \(a\) is the first term, \(r\) is the common ratio, and \(n\) represents the number of terms added together in the sum.

Using this formula, it becomes straightforward to calculate sums even for large \(n\). Remember to make sure the common ratio \(r\) is not 1, as the formula requires dividing by \(1-r\).

Common Ratio

The common ratio is a key characteristic of a geometric sequence, defined by the factor you multiply each term by to get the next term. It is denoted by \(r\). For instance, if you have a sequence where each term is one-third of the previous one, the common ratio is \(\frac{1}{3}\).

Here's how to determine it:

• Take any term in the sequence and divide it by the previous term. For example: second term divided by the first term.
• In our exercise: \(r = \frac{1}{3}\).

Understanding the common ratio helps in determining the nature and direction (increasing or decreasing) of the sequence. A common ratio greater than 1 implies the sequence is growing, while less than 1 shows it's decreasing.

Formula for Geometric Sequence

A geometric sequence is defined by its first term and the common ratio. It follows a specific formula to find any term in the sequence. The general term of a geometric sequence can be calculated using:

• \(a_n = a \cdot r^{(n-1)}\)

Where:

• \(a_n\) is the \(n\)-th term,
• \(a\) is the first term,
• \(r\) is the common ratio, and
• \(n\) is the term number you wish to find.

In cases where the number of terms \(n\) is given, like in our exercise, it's possible to substitute \(n\) into both the geometric sequence formula and the partial sum formula to find specific values.

First Term

The first term of a geometric sequence is crucial as it sets the base for all following terms. Denoted by \(a\), it is the starting point from which all subsequent terms are calculated by multiplying by the common ratio \(r\).

In any sequence, knowing \(a\) allows you to understand the pattern of the sequence deeply. For example, if \(a = \frac{2}{3}\), this initial value guides the progression and behavior of the sequence.

The first term is also vital when using formulas to find not only specific terms but also the sum of terms within a sequence. Without \(a\), calculations of both the \(n\)-th term and partial sums become impossible.