Problem 35 - Precalculus Mathematics for Calculus

Question

Find the first six partial sums \(S_{1}, S_{2}, S_{3}, S_{4}, S_{5}, S_{6}\) of the sequence. $$\frac{1}{3}, \frac{1}{3^{2}}, \frac{1}{3^{3}}, \frac{1}{3^{4}}, \ldots$$

Step-by-Step Solution

Answer

The first six partial sums are \(\frac{1}{3}, \frac{4}{9}, \frac{13}{27}, \frac{40}{81}, \frac{121}{243}, \frac{364}{729}\).

Step 1: Identify the Sequence

Recognize that the given sequence is geometric with the first term \(a = \frac{1}{3}\) and common ratio \(r = \frac{1}{3}\).

Step 2: Calculate the First Partial Sum \(S_{1}\)

Calculate \(S_{1}=\frac{1}{3}\) since it is the sum of the first term.

Step 3: Calculate the Second Partial Sum \(S_{2}\)

Sum the first two terms: \(S_{2} = \frac{1}{3} + \frac{1}{3^2} = \frac{1}{3} + \frac{1}{9} = \frac{3}{9} + \frac{1}{9} = \frac{4}{9}\).

Step 4: Calculate the Third Partial Sum \(S_{3}\)

Sum the first three terms: \(S_{3} = \frac{1}{3} + \frac{1}{9} + \frac{1}{27} = \frac{4}{9} + \frac{1}{27} = \frac{12}{27} + \frac{1}{27} = \frac{13}{27}\).

Step 5: Calculate the Fourth Partial Sum \(S_{4}\)

Sum the first four terms: \(S_{4} = \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \frac{1}{81} = \frac{13}{27} + \frac{1}{81} = \frac{39}{81} + \frac{1}{81} = \frac{40}{81}\).

Step 6: Calculate the Fifth Partial Sum \(S_{5}\)

Sum the first five terms: \(S_{5} = \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \frac{1}{81} + \frac{1}{243} = \frac{40}{81} + \frac{1}{243} = \frac{120}{243} + \frac{1}{243} = \frac{121}{243}\).

Step 7: Calculate the Sixth Partial Sum \(S_{6}\)

Sum the first six terms: \(S_{6} = \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \frac{1}{81} + \frac{1}{243} + \frac{1}{729} = \frac{121}{243} + \frac{1}{729} = \frac{363}{729} + \frac{1}{729} = \frac{364}{729}\). Simplify if possible.

Key Concepts

Partial Sums in a Geometric Series

The concept of partial sums is central to understanding geometric series. A partial sum, represented by \(S_n\), is the sum of the first \(n\) terms of a sequence. In a geometric series, where each term after the first is found by multiplying the previous term by a constant called the 'common ratio', calculating partial sums is straightforward once the common ratio is identified.

Consider a geometric sequence with the first term \(a\) and the common ratio \(r\). The partial sum of the first \(n\) terms is given by the formula:

• \(S_n = a + ar + ar^2 + \ldots + ar^{n-1}\)
• There is also a formula: \(S_n = a \frac{1-r^n}{1-r}\) if \(|r| < 1\)

For instance, in the provided exercise, if you calculate the first six terms' partial sums using this knowledge, students deepen their understanding of how each subsequent partial sum is built incrementally upon the previous ones. The process involves progressively adding each new term of the sequence to the cumulative total of the preceding terms.

Understanding Sequences

A sequence is an ordered list of numbers, and in mathematics, we often work with different types of sequences. Geometric sequences are specific types where each number, after the first, is the product of the previous number and a fixed non-zero number called the 'common ratio'.

For example, the sequence in the exercise is a perfect example of a geometric sequence: \(\frac{1}{3}, \frac{1}{3^{2}}, \frac{1}{3^{3}}, \frac{1}{3^{4}}, \ldots\)
Here, every term is obtained by multiplying the previous term by \(\frac{1}{3}\).
These repetitive processes create a predictable pattern, making it easier to anticipate any term's value in the sequence given the position, or vice versa. Studying sequences helps reveal patterns and provides a foundation for more complex mathematical concepts.

The Role of the Common Ratio

In a geometric sequence, the common ratio \(r\) is the factor that dictates the change from one term to the next. It is essential for understanding how a sequence develops over time.

The common ratio is calculated by dividing any term in the sequence by its preceding term. For example, in our series \(\frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \ldots\), every term is the previous term multiplied by \(\frac{1}{3}\). This consistent relationship defines the rate of progression in the sequence.

Moreover, the value of \(r\) also helps identify the behavior of the sequence over time:

• If \(|r| < 1\), the terms in the sequence will get smaller and approach zero.
• If \(|r| = 1\), the terms remain the same.
• If \(|r| > 1\), the terms grow larger, rapidly increasing in size.

Understanding the common ratio allows students to predict the nature of the sequence and calculate further terms or partial sums efficiently.