Problem 13 - Algebra for College Students

Question

List all terms of each finite sequence. \(a_{n}=2^{-n}\) for \(1 \leq n \leq 6\)

Step-by-Step Solution

Answer

The terms are \(\frac{1}{2}\), \(\frac{1}{4}\), \(\frac{1}{8}\), \(\frac{1}{16}\), \(\frac{1}{32}\), and \(\frac{1}{64}\).

Step 1: Understand the Sequence

The sequence is given by the formula \(a_{n} = 2^{-n}\). We need to find the first 6 terms where \(n\) ranges from 1 to 6.

Step 2: Calculate the First Term (n=1)

Substitute \(n = 1\) into the formula: \(a_{1} = 2^{-1} = \frac{1}{2}\).

Step 3: Calculate the Second Term (n=2)

Substitute \(n = 2\) into the formula: \(a_{2} = 2^{-2} = \frac{1}{4}\).

Step 4: Calculate the Third Term (n=3)

Substitute \(n = 3\) into the formula: \(a_{3} = 2^{-3} = \frac{1}{8}\).

Step 5: Calculate the Fourth Term (n=4)

Substitute \(n = 4\) into the formula: \(a_{4} = 2^{-4} = \frac{1}{16}\).

Step 6: Calculate the Fifth Term (n=5)

Substitute \(n = 5\) into the formula: \(a_{5} = 2^{-5} = \frac{1}{32}\).

Step 7: Calculate the Sixth Term (n=6)

Substitute \(n = 6\) into the formula: \(a_{6} = 2^{-6} = \frac{1}{64}\).

Step 8: List All Terms

List the terms calculated: \(a_{1} = \frac{1}{2}\), \(a_{2} = \frac{1}{4}\), \(a_{3} = \frac{1}{8}\), \(a_{4} = \frac{1}{16}\), \(a_{5} = \frac{1}{32}\), \(a_{6} = \frac{1}{64}\).

Key Concepts

Sequence Formula

A sequence formula is a mathematical expression that defines the terms within a sequence. In this case, our sequence is defined by the formula:

\(a_{n} = 2^{-n}\)

This formula tells us how to calculate the value for any term in the sequence by substituting the term position (\(n\)) into the formula. For instance, if we want to find the term when \(n = 3\), we substitute 3 into the sequence formula, resulting in:

\(a_{3} = 2^{-3} = \frac{1}{8}\)

Formulas like these make it easier to understand the structure of sequences and quickly find specific terms. They are crucial in simplifying otherwise complex calculations.

Exponential Sequence

An exponential sequence is a type of sequence where each term is a result of raising a fixed base to varying exponents. In our example, the base is 2, and the exponent is the negative of the term index (\( -n \)).

Exponential sequences demonstrate dramatic changes in values, whether growing or shrinking. In our sequence:

\[a_{n} = 2^{-n}
\]

Each successive term becomes half the value of the previous one. Here's a breakdown of the first few terms:

• \(a_{1} = 2^{-1} = \frac{1}{2}\)
• \(a_{2} = 2^{-2} = \frac{1}{4}\)
• \(a_{3} = 2^{-3} = \frac{1}{8}\)
• \(a_{4} = 2^{-4} = \frac{1}{16}\)
• \(a_{5} = 2^{-5} = \frac{1}{32}\)
• \(a_{6} = 2^{-6} = \frac{1}{64}\)

By understanding the nature of an exponential sequence, students can predict how quick the values decrease or increase, hence understanding the overall pattern.

Term Calculation

Term calculation involves substituting the term's position (denoted as \(n\)) into the sequence formula to find its value. Let's break down the process for our sequence step-by-step:

1. Understand the Sequence Formula:
Since we have \(a_{n} = 2^{-n}\), we see that each term is found by raising 2 to the power of the negative term index.

2. Calculate each Term:

• For \(n = 1\), substitute into the formula: \(a_{1} = 2^{-1} = \frac{1}{2}\).
• For \(n = 2\), substitute into the formula: \(a_{2} = 2^{-2} = \frac{1}{4}\).
• For \(n = 3\), substitute into the formula: \(a_{3} = 2^{-3} = \frac{1}{8}\).
• For \(n = 4\), substitute into the formula: \(a_{4} = 2^{-4} = \frac{1}{16}\).
• For \(n = 5\), substitute into the formula: \(a_{5} = 2^{-5} = \frac{1}{32}\).
• For \(n = 6\), substitute into the formula: \(a_{6} = 2^{-6} = \frac{1}{64}\).

3. List the Terms:
After calculating all values, we can list the sequence terms: \(\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \frac{1}{32}, \frac{1}{64}\). Breaking down the terms like this makes it easier to understand how each value in the sequence is derived.