Problem 77 - Algebra and Trigonometry

Question

Determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. If the sequence is arithmetic or geometric, find the sum of the first 50 terms. $$ \left\\{\left(\frac{2}{3}\right)^{n}\right\\} $$

Step-by-Step Solution

Answer

The sequence is geometric with a common ratio of \(\frac{2}{3}\). The sum of the first 50 terms is \(2 \left(1-\left(\frac{2}{3}\right)^{50}\right)\).

Step 1: Identify the type of sequence

Examine the given sequence \(\left\{\left(\frac{2}{3}\right)^{n}\right\}\). This sequence is of the form \(a^{n}\), which suggests it might be a geometric sequence.

Step 2: Determine the common ratio

For a geometric sequence, the common ratio \(r\) can be found by dividing any term by its preceding term. Consider the first two terms:\ \(a_{1} = \left(\frac{2}{3}\right)^{1} = \frac{2}{3}\) and \(a_{2} = \left(\frac{2}{3}\right)^{2} = \left(\frac{2}{3}\right) \cdot \left(\frac{2}{3}\right) = \frac{4}{9}\).\ \ \(r = \frac{a_{2}}{a_{1}} = \frac{\frac{4}{9}}{\frac{2}{3}} = \frac{4}{9} \cdot \frac{3}{2} = \frac{4}{6} = \frac{2}{3}\). Thus, the common ratio is \(\frac{2}{3}\).

Step 3: Verify that the sequence is geometric

To confirm that the sequence is geometric, ensure that this ratio holds for other consecutive terms. For example, check \(a_{3} = \left(\frac{2}{3}\right)^{3} = \frac{8}{27}\) and verify that \(\frac{a_{3}}{a_{2}} = \frac{\frac{8}{27}}{\frac{4}{9}} = \frac{8}{27} \cdot \frac{9}{4} = \frac{8}{3} \cdot \frac{1}{3} = \frac{8}{12} = \frac{2}{3}\). Hence, the sequence is confirmed as geometric.

Step 4: Sum of the first 50 terms

For a geometric sequence, the sum of the first \(n\) terms is given by \[S_{n} = a \cdot \frac{1-r^{n}}{1-r}\]. Here, \(a = \frac{2}{3}\) and \(r = \frac{2}{3}\), and we want the sum of the first 50 terms (n = 50).\ Substitute the values: \[S_{50} = \frac{2}{3} \cdot \frac{1-\left(\frac{2}{3}\right)^{50}}{1-\frac{2}{3}} = \frac{2}{3} \cdot \frac{1-\left(\frac{2}{3}\right)^{50}}{\frac{1}{3}} = 2 \cdot \left(1-\left(\frac{2}{3}\right)^{50}\right)= 2 \left(1-\left(\frac{2}{3}\right)^{50}\right)\]

Key Concepts

common ratio

The common ratio is an essential concept in geometric sequences. A geometric sequence is defined by a constant ratio between consecutive terms. This ratio is known as the common ratio and is denoted by the symbol \(r\).

To find the common ratio in a geometric sequence, you take any term and divide it by the preceding term. For instance, consider the initial terms of the given sequence \(\frac{2}{3}, \frac{4}{9}, \frac{8}{27}\). To find the common ratio, compute the following:

- \(r = \frac{a_{2}}{a_{1}} = \frac{\frac{4}{9}}{\frac{2}{3}} = \frac{4}{9} \times \frac{3}{2} = \frac{4}{6} = \frac{2}{3}\)

This calculation shows that the common ratio \(r\) for our sequence is \( \frac{2}{3}\). The common ratio helps us understand how the sequence grows or shrinks and is crucial for other calculations like finding the sum of terms.

sequence and series

Sequences and series form fundamental concepts in algebra and calculus. A sequence is a list of numbers arranged in a specific order according to a given rule. In our case, the sequence is geometric, where each term is obtained by multiplying the previous term by a constant ratio, \(\frac{2}{3}\).

Series, on the other hand, is the sum of the terms of a sequence. For a geometric sequence, the series can be finite or infinite. Understanding the difference between these two is crucial as they have different formulas and properties. For example, our exercise involves summing the first 50 terms, a finite geometric series.

The power of understanding sequences and series lies in various applications: from simple interest calculations to more complex financial modeling and even nature's growth patterns.

sum of terms

Calculating the sum of the terms of a sequence is a common task in mathematics. For geometric sequences like \(\frac{2}{3}^{n}\), this involves a specific formula:

\( S_{n} = a \times \frac{1-r^{n}}{1-r} \)

Where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms. Let's apply this formula to find the sum of the first 50 terms of our sequence, where \(a = \frac{2}{3}\) and \(r = \frac{2}{3}\).

Substitute in the values:

- \( S_{50} = \frac{2}{3} \times \frac{1-\big(\frac{2}{3}\big)^{50}}{1-\frac{2}{3}} = \frac{2}{3} \times \frac{1-\big(\frac{2}{3}\big)^{50}}{\frac{1}{3}} = 2 \times [1-\big(\frac{2}{3}\big)^{50}] = 2 \times [1-(\frac{2}{3})^{50}] \)

This simplified result allows us to understand how sums are calculated for geometric sequences. Understanding and applying such formulas helps in diverse areas like physics, economics, and computer science.