Problem 21 - Algebra and Trigonometry

Question

List the first five terms of each sequence. \(\left\\{s_{n}\right\\}=\left\\{\frac{3^{n}}{2^{n}+3}\right\\}\)

Step-by-Step Solution

Answer

\(s_1 = \frac{3}{5}, s_2 = \frac{9}{7}, s_3 = \frac{27}{11}, s_4 = \frac{81}{19}, s_5 = \frac{243}{35}\)

Step 1: Identify the first term (n = 1)

Substitute \(n = 1\) into the formula: \[ s_1 = \frac{3^1}{2^1 + 3} = \frac{3}{2 + 3} = \frac{3}{5} \]

Step 2: Calculate the second term (n = 2)

Substitute \(n = 2\) into the formula: \[ s_2 = \frac{3^2}{2^2 + 3} = \frac{9}{4 + 3} = \frac{9}{7} \]

Step 3: Calculate the third term (n = 3)

Substitute \(n = 3\) into the formula: \[ s_3 = \frac{3^3}{2^3 + 3} = \frac{27}{8 + 3} = \frac{27}{11} \]

Step 4: Calculate the fourth term (n = 4)

Substitute \(n = 4\) into the formula: \[ s_4 = \frac{3^4}{2^4 + 3} = \frac{81}{16 + 3} = \frac{81}{19} \]

Step 5: Calculate the fifth term (n = 5)

Substitute \(n = 5\) into the formula: \[ s_5 = \frac{3^5}{2^5 + 3} = \frac{243}{32 + 3} = \frac{243}{35} \]

Key Concepts

sequence calculation

Understanding sequences is a key topic in algebra. A sequence is an ordered list of numbers, where each number is called a term. To calculate terms within a sequence, you usually insert values into a given formula.
For this exercise, a sequence is given by the formula \(\frac{3^{n}}{2^{n}+3}\). Let's break it down:
1. For each term, substitute the term number \( n \) into the formula.
2. Calculate the numerator \( 3^n \) and the denominator \( 2^n + 3 \), then divide them.
Recall: A sequence is unique because even though the formula stays the same, each term is different due to the change in \( n \).
Listing the first five terms gives us:

• The first term \( s_1 = \frac{3}{5} \)
• The second term \( s_2 = \frac{9}{7} \)
• The third term \( s_3 = \frac{27}{11} \)
• The fourth term \( s_4 = \frac{81}{19} \)
• The fifth term \( s_5 = \frac{243}{35} \)

This approach helps in understanding the pattern within sequences and their behavior as \( n \) increases.

exponential functions

Exponential functions play a crucial role in various sequences. An exponential function has the form \( a^x \), where \( a \) (base) is a constant and \( x \) is the exponent. In our sequence formula, \( 3^n \) is an example of an exponential function.
Exponential functions grow or decay very quickly:

• When the base \( a > 1 \), the function grows rapidly.
• When \( 0 < a < 1 \), the function decays (shrinks) rapidly.

Key components to understand:

• Base (3 in \( 3^n \)): determines the rate of growth. Here it grows by a factor of 3 each term.
• Exponent (n): determines the term in the sequence, increasing progressively as \( n \) increases.

For example, \( 3^2 = 9 \), \( 3^3 = 27 \), illustrating rapid growth as \( n \) increases. Understanding these rapid changes helps in predicting sequence patterns and managing exponential growth effectively.

rational expressions

Rational expressions are fractions where the numerator and/or denominator are polynomials. They appear frequently in sequences and other algebraic structures.
In our sequence, \(\frac{3^n}{2^n + 3}\) is a rational expression.
Breaking it down:

• The numerator \( 3^n \) is a polynomial in the form of an exponential function.
• The denominator \( 2^n + 3 \) is another polynomial.

This type of expression can be simplified, but simplification may depend on specific values of \( n \). For each term:

• Evaluate the exponent in the numerator.
• Add and evaluate the polynomial in the denominator.
• Divide them to get a rational expression.

For instance, in \( s_3 = \frac{27}{11} \), \( 27 \) is from \( 3^3 \) and \( 11 \) is from the sum of \( 2^3 + 3 \). Rational expressions simplify complex calculations, making it easier to understand terms in sequences.