Problem 38 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the indicated term of the sequence. $$\begin{aligned} &a_{n}=\frac{3^{n}}{3^{n}+1}\\\ &a_{6}= \end{aligned}$$

Step-by-Step Solution

Answer

The 6th term of the sequence is \(\frac{729}{730}\).

Step 1: Understand the sequence formula

The general form of the sequence given is \(a_{n}=\frac{3^{n}}{3^{n}+1}\), where \(a_{n}\) is the \(n\)th term of the sequence.

Step 2: Substitute n with the term we need to find

The problem asks for the 6th term of the sequence, so set \(n=6\) and substitute it into the formula: \(a_{6}=\frac{3^{6}}{3^{6}+1}\).

Step 3: Evaluate the expression for the 6th term

First, calculate the value of \(3^{6}\) which is \(729\). Then substitute that value back into the equation: \(a_{6}=\frac{729}{729+1} = \frac{729}{730}\).

Key Concepts

Understanding the Sequence Formula

In mathematics, a sequence is a set of numbers arranged in a specific order. Each number in a sequence is called a term. The formula of a sequence helps us determine any term in that sequence. For instance, the sequence given by the formula \(a_{n} = \frac{3^{n}}{3^{n} + 1}\) helps us find any term's value by simply plugging in the desired term's position, \(n\).
In our example, this formula represents each term as a ratio of exponential expressions. The numerator and the denominator both have the expression \(3^{n}\), making it crucial for understanding the sequence's behavior.
Understanding the sequence formula is key to evaluating or predicting the sequence's future terms. This includes knowing how to handle exponents and basic algebraic manipulations.

Techniques for Evaluating Expressions

Evaluating an expression means calculating its value. In the context of sequences, it's about finding the specific value of a term. Here's how it works:
1. **Substitution:** Start by substituting the given value of \(n\) into your sequence formula.
2. **Simplification:** Solve the expression using arithmetic operations like addition or multiplication. In sequences, this often involves dealing with powers or multiplying terms.
3. **Final Calculation:** Continue simplifying until you reach the final form, which is often a single fraction or number. This step is crucial because it gives the exact term's value in the sequence.
For instance, substituting \(n = 6\) in our sequence formula, and simplifying, involves calculating \(3^{6}\) and finally reducing the expression to \(\frac{729}{730}\). This simplification directly provides us with the sixth term.

Working with Exponential Expressions

Exponential expressions involve numbers raised to a certain power and are a fundamental part of sequences like the one we're working with. In our sequence formula, the terms include \(3^{n}\), which requires understanding exponents.
Some important points to remember:

• **Power Calculation:** An exponential expression like \(3^{6}\) means multiplying 3 by itself 6 times, giving 729.
• **Combination with Fractions:** These powers often appear in fractions, requiring careful addition or subtraction for combining with other numbers, as seen when calculating \(3^{6} + 1\).

Understanding how to handle these exponential expressions is crucial for evaluating sequence terms, ensuring you correctly find the numerator and denominator before simplification.