Problem 22 - Algebra and Trigonometry

Question

List the first five terms of each sequence. \(\left\\{s_{n}\right\\}=\left\\{\left(\frac{4}{3}\right)^{n}\right\\}\)

Step-by-Step Solution

Answer

The first five terms are: \(\frac{4}{3}\), \(\frac{16}{9}\), \(\frac{64}{27}\), \(\frac{256}{81}\), and \(\frac{1024}{243}\).

Step 1 - Understand the Sequence Definition

The sequence is given by \{s_{n}\} = \{(\frac{4}{3})^{n}\}. This means each term of the sequence is calculated by raising \(\frac{4}{3}\) to the power of \(n\) (where \(n\) is the term number).

Step 2 - Calculate the First Term

The first term corresponds to \(n = 1\). Calculate \(s_{1}\) by substituting \(n = 1\) into the sequence formula: \(s_{1} = (\frac{4}{3})^{1} = \frac{4}{3}\).

Step 3 - Calculate the Second Term

The second term corresponds to \(n = 2\). Calculate \(s_{2}\) by substituting \(n = 2\) into the sequence formula: \(s_{2} = (\frac{4}{3})^{2} = \frac{16}{9}\).

Step 4 - Calculate the Third Term

The third term corresponds to \(n = 3\). Calculate \(s_{3}\) by substituting \(n = 3\) into the sequence formula: \(s_{3} = (\frac{4}{3})^{3} = \frac{64}{27}\).

Step 5 - Calculate the Fourth Term

The fourth term corresponds to \(n = 4\). Calculate \{s_{4)\} by substituting \(n = 4\) into the sequence formula: \(s_{4} = (\frac{4}{3})^{4} = \frac{256}{81}\).

Step 6 - Calculate the Fifth Term

The fifth term corresponds to \(n = 5\). Calculate \(s_{5}\) by substituting \(n = 5\) into the sequence formula: \(s_{5} = (\frac{4}{3})^{5} = \frac{1024}{243}\).

Key Concepts

Sequence Terms Calculation

A sequence is a set of numbers arranged in a specific order. To find terms in a geometric sequence, you use a formula that relates to each term's position in the sequence. For the sequence given by \(\left\{s_{n}\right\}\ = \left\{\left(\frac{4}{3}\right)^{n}\right\}\), each term is calculated by raising the base value, \(\frac{4}{3}\), to the power of the term's position index.
Calculating each term involves substituting the term index (e.g., 1, 2, 3, etc.) into the formula. For example:

• Calculate the first term (\(n=1\)): \(s_{1} = (\frac{4}{3})^{1} = \frac{4}{3}\)
• Calculate the second term (\(n=2\)): \(s_{2} = (\frac{4}{3})^{2} = \frac{16}{9}\)
• Calculate the third term (\(n=3\)): \(s_{3} = (\frac{4}{3})^{3} = \frac{64}{27}\)
• Calculate the fourth term (\(n=4\)): \(s_{4} = (\frac{4}{3})^{4} = \frac{256}{81}\)
• Calculate the fifth term (\(n=5\)): \(s_{5} = (\frac{4}{3})^{5} = \frac{1024}{243}\)

Understanding how to substitute and calculate the terms allows you to find any term in the sequence.

Exponential Sequences

Exponential sequences are sequences where each term is found by multiplying the previous term by a constant. In our example sequence \(\left\{(\frac{4}{3})^{n}\right\}\), the constant multiplier is \(\frac{4}{3}\). This means with each step, we multiply the previous term by \(\frac{4}{3}\).
Exponential sequences grow very quickly because they are based on repeated multiplication.

• For \(n=2\), \(s_{2} = s_{1} \cdot \frac{4}{3} = \frac{4}{3} \cdot \frac{4}{3} = \frac{16}{9}\)
• For \(n=3\), \(s_{3} = s_{2} \cdot \frac{4}{3} = \frac{16}{9} \cdot \frac{4}{3} = \frac{64}{27}\)
• For \(n=4\), \(s_{4} = s_{3} \cdot \frac{4}{3} = \frac{64}{27} \cdot \frac{4}{3} = \frac{256}{81}\)
• For \(n=5\), \(s_{5} = s_{4} \cdot \frac{4}{3} = \frac{256}{81} \cdot \frac{4}{3} = \frac{1024}{243}\)

Algebraic Expressions

In mathematics, an algebraic expression is a combination of numbers, variables, and operations (such as addition, multiplication, etc.). In the context of our sequence \(\left\{(\frac{4}{3})^{n}\right\}\), the formula \( (\frac{4}{3})^{n} \) is an algebraic expression. Here, \((\frac{4}{3})^{n}\) combines division, exponentiation, and the variable \(n\). Understanding algebraic expressions is important because they form the basis of many mathematical sequences and functions. Breaking them down into simpler components helps grasp their behavior and apply them in various problems.

• The numerator in \(\frac{4}{3}\) tells you the base number being repeatedly multiplied.
• The denominator shows how each term shrinks or grows, combining into an exponential growth pattern.
• The exponent \(n\) indicates the term's position in the sequence.