Problem 37 - Precalculus Mathematics for Calculus

Question

Find the first four partial sums and the \(n\) th partial sum of the sequence \(a_{n^{*}}\) $$a_{n}=\frac{2}{3^{n}}$$

Step-by-Step Solution

Answer

The first four partial sums are \(\frac{2}{3}, \frac{8}{9}, \frac{26}{27}, \frac{80}{81}\) and the n-th partial sum is \(2\left(1 - \frac{1}{3^n}\right)\).

Step 1: Understand the Sequence

The sequence given is \(a_n = \frac{2}{3^n}\). This sequence represents a series where each term is given by \(a_n\). Here, \(n\) is the term number in the sequence.

Step 2: Calculate the First Term

The first term of the sequence \(S_1\) is simply \(a_1\), which is calculated as \[ a_1 = \frac{2}{3^1} = \frac{2}{3}. \] Thus, the first partial sum \(S_1\) is \(\frac{2}{3}\).

Step 3: Calculate the Second Partial Sum

To find the second partial sum \(S_2\), we add the first two terms of the sequence: \[ S_2 = a_1 + a_2 = \frac{2}{3} + \frac{2}{3^2} = \frac{2}{3} + \frac{2}{9}. \] Converting \(\frac{2}{3}\) to a common denominator with \(\frac{2}{9}\), we have \(\frac{6}{9}\). Thus, \[ S_2 = \frac{6}{9} + \frac{2}{9} = \frac{8}{9}. \]

Step 4: Calculate the Third Partial Sum

The third partial sum \(S_3\) is the sum of the first three terms: \[ S_3 = a_1 + a_2 + a_3 = \frac{2}{3} + \frac{2}{9} + \frac{2}{27}. \] Converting all terms to have a common denominator (27), we get \(\frac{18}{27} + \frac{6}{27} + \frac{2}{27}\). Therefore, \[ S_3 = \frac{18+6+2}{27} = \frac{26}{27}. \]

Step 5: Calculate the Fourth Partial Sum

The fourth partial sum \(S_4\) is the sum of the first four terms: \[ S_4 = a_1 + a_2 + a_3 + a_4 = \frac{2}{3} + \frac{2}{9} + \frac{2}{27} + \frac{2}{81}. \] Converting all terms to have a common denominator (81), we get \(\frac{54}{81} + \frac{18}{81} + \frac{6}{81} + \frac{2}{81}\). Thus, \[ S_4 = \frac{54+18+6+2}{81} = \frac{80}{81}. \]

Step 6: Define the General Formula for the n-th Partial Sum

The partial sums form a geometric series. The formula for the n-th partial sum \(S_n\) of a geometric series where \(a\) is the first term and \(r\) is the common ratio (where \(|r| < 1\)) is given by \[ S_n = a\frac{1 - r^n}{1 - r}. \] Here, \(a = \frac{2}{3}\) and \(r = \frac{1}{3}\), so the n-th partial sum is \[ S_n = \frac{2}{3}\frac{1 - \left(\frac{1}{3}\right)^n}{1 - \frac{1}{3}}. \] This simplifies to \[ S_n = 2\left(1 - \frac{1}{3^n}\right). \]

Key Concepts

Geometric Sequence

A geometric sequence, or geometric progression, is an ordered set of numbers where each term after the first is found by multiplying the previous one by a constant called the 'common ratio' (denoted as \(r\)). The formula for a basic geometric sequence can be written as \(a_n = a_1 \, r^{n-1}\), where \(a_1\) is the first term and \(n\) indicates the position of a term within the sequence. In the sequence \(a_n = \frac{2}{3^n}\), every new term is formed by multiplying the previous term by the common ratio \(\frac{1}{3}\), starting with the first term at \(n=1\). This characteristic makes it easier to predict future terms in the sequence by utilizing the properties of exponents combined with the initial term and common ratio.

Geometric Series

A geometric series is the sum of the terms of a geometric sequence. In mathematics, a series represents a sum of sequential terms. When specifically referring to geometric sequences, a geometric series takes the form of a sum like \(S = a + ar + ar^2 + ar^3 + \ldots\). In the provided solution, the partial sums (\(S_1, S_2, S_3,\) etc.) depict the sums of the initial terms of the geometric sequence \(a_n = \frac{2}{3^n}\). The concept of partial sums is pivotal because it lets us see how the sum converges towards a specific limit, particularly when the common ratio is a fraction between \(-1\) and \(1\). In this case, the series converges because each additional term added gets smaller and smaller, thanks to the divisibility by ever-increasing powers of 3.

n-th Term Formula

The n-th term formula, specific to geometric sequences, helps find the value of a term at any position \(n\) without listing all previous terms. The general formula to find the n-th term \(a_n\) of a geometric sequence is given by:

• Formula: \(a_n = a_1 \cdot r^{n-1}\)
• Where: \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the term number.

For the n-th partial sum \(S_n\) of a geometric series, the formula changes slightly, considering the sum of terms:

• Formula for Partial Sum: \(S_n = a \frac{1 - r^n}{1 - r}\) if \(|r| < 1\)
• In our sequence: \(S_n = \frac{2}{3} \frac{1 - (\frac{1}{3})^n}{1 - \frac{1}{3}} = 2(1 - \frac{1}{3^n})\)

This formula is useful for quickly deriving the sum of the first \(n\) terms in the series, especially when the series converges as \(n\) approaches infinity.