Problem 37 - Algebra and Trigonometry

Question

\(37-40\) . Find the first four partial sums and the nth partial sum of the sequence \(a_{m}\) $$ a_{n}=\frac{2}{3^{n}} $$

Step-by-Step Solution

Answer

The first four partial sums are \(\frac{2}{3}\), \(\frac{8}{9}\), \(\frac{26}{27}\), and \(\frac{240}{243}\). The nth partial sum is \(2\left(1 - \left(\frac{1}{3}\right)^n\right)\).

Step 1: Understanding the Sequence

The sequence given is a geometric sequence where each term is of the form \(a_{n} = \frac{2}{3^n}\). This means that the first term is \(a_{1} = \frac{2}{3}\), the second term is \(a_{2} = \frac{2}{9}\), and so forth.

Step 2: Calculate the First Partial Sum

The first partial sum \(S_1\) is just the first term of the sequence: \[ S_1 = a_1 = \frac{2}{3} \].

Step 3: Calculate the Second Partial Sum

The second partial sum \(S_2\) includes the first two terms of the sequence: \[ S_2 = a_1 + a_2 = \frac{2}{3} + \frac{2}{9} \]. To add these, find a common denominator and simplify: \[ S_2 = \frac{6}{9} + \frac{2}{9} = \frac{8}{9} \].

Step 4: Calculate the Third Partial Sum

The third partial sum \(S_3\) adds the first three terms of the sequence: \[ S_3 = a_1 + a_2 + a_3 = \frac{2}{3} + \frac{2}{9} + \frac{2}{27} \]. Find a common denominator: \[ S_3 = \frac{18}{27} + \frac{6}{27} + \frac{2}{27} = \frac{26}{27} \].

Step 5: Calculate the Fourth Partial Sum

The fourth partial sum \(S_4\) includes the first four terms: \[ S_4 = a_1 + a_2 + a_3 + a_4 = \frac{2}{3} + \frac{2}{9} + \frac{2}{27} + \frac{2}{81} \]. Use a common denominator: \[ S_4 = \frac{162}{243} + \frac{54}{243} + \frac{18}{243} + \frac{6}{243} = \frac{240}{243} \].

Step 6: Calculate the nth Partial Sum

The nth partial sum of a geometric series \(a_{n} = ar^{n-1}\) with first term \(a\) and common ratio \(r\) can be given by the formula: \[ S_n = a \frac{1-r^n}{1-r} \]. For this sequence where \(a = \frac{2}{3}\) and \(r = \frac{1}{3}\): \[ S_n = \frac{2}{3} \frac{1 - \left(\frac{1}{3}\right)^n}{1 - \frac{1}{3}} = 2 \left(1 - \left(\frac{1}{3}\right)^n\right) \].

Key Concepts

Partial Sums

In a geometric sequence like the one presented, finding partial sums means calculating the sum of a subset of the sequence's terms, starting from the first. Partial sums are crucial as they help understand how the sequence behaves over time by observing cumulative totals.
Let's break down the concept step-by-step:

• The first partial sum, denoted as \( S_1 \), comprises only the first term of the sequence. It represents the starting point.
• The second partial sum, \( S_2 \), is the sum of the first two terms, showing an accumulation pattern.
• As we proceed, \( S_3 \), the third partial sum, accumulates the first three terms, providing a deeper insight into the growth or shrinkage of the sequence.
• Lastly, \( S_4 \), includes up to the fourth term, further building on the summed values.

The power of understanding partial sums lies in visualizing the sequence’s progression, crucial for grasping complex concepts like convergence or divergence in sequences and series.

Geometric Series Formula

The geometric series formula is indispensable when it comes to summing up terms in a geometric sequence. A geometric sequence is defined by a constant ratio between consecutive terms. This consistency allows us to use a specific formula to find the sum of the series up to a certain number of terms.

A geometric series' sum can be calculated using the formula:

\[ S_n = a \frac{1-r^n}{1-r} \]

Here,

• \( S_n \) is the sum of the first \( n \) terms.
• \( a \) stands for the first term of the sequence.
• \( r \) is the common ratio (how much you multiply each term by to get the next one).

The beauty of this formula lies in its simplicity and effectiveness, allowing us to quickly find the sum of any number of terms in a geometric sequence, without calculating each term individually. By understanding this formula, you unlock the ability to analyze sequence patterns succinctly.

Nth Term Calculation

Finding the nth term in a geometric sequence is essential for understanding and predicting the behavior of the sequence over time. The formula for the nth term, \( a_{n} \), is derived from the pattern that each term is formed by multiplying the previous term by a constant ratio, \( r \).

The formula for the nth term in a geometric sequence is:

\[ a_n = a \cdot r^{n-1} \]

Where:

• \( a_n \) is the nth term we want to find.
• \( a \) is the first term in the sequence.
• \( r \) is the common ratio between consecutive terms.
• \( n \) is the position of the term in the sequence.

This formula gives you a straightforward way to calculate any term in the sequence. With it, you can explore hypothetical scenarios by predicting future terms or confirming the position of any given term in the sequence.