Problem 5 - Thomas Calculus

Question

Each of Exercises \(1-6\) gives a formula for the \(n\) th term \(a_{n}\) of a sequence \(\left\\{a_{n}\right\\} .\) Find the values of \(a_{1}, a_{2}, a_{3},\) and \(a_{4} .\) $$ a_{n}=\frac{2^{n}}{2^{n+1}} $$

Step-by-Step Solution

Answer

All terms \(a_1, a_2, a_3,\) and \(a_4\) equal \(\frac{1}{2}\).

Step 1: Identify the General Formula

The formula for the sequence is given as \(a_n = \frac{2^n}{2^{n+1}}\). This represents the general term of the sequence. Our task is to find specific terms: \(a_1\), \(a_2\), \(a_3\), and \(a_4\).

Step 2: Calculate the First Term

Substitute \(n = 1\) into the general formula to find \(a_1\):\[a_1 = \frac{2^1}{2^{1+1}} = \frac{2}{2^2} = \frac{2}{4} = \frac{1}{2}\]

Step 3: Calculate the Second Term

Substitute \(n = 2\) into the general formula to find \(a_2\):\[a_2 = \frac{2^2}{2^{2+1}} = \frac{4}{2^3} = \frac{4}{8} = \frac{1}{2}\]

Step 4: Calculate the Third Term

Substitute \(n = 3\) into the general formula to find \(a_3\):\[a_3 = \frac{2^3}{2^{3+1}} = \frac{8}{2^4} = \frac{8}{16} = \frac{1}{2}\]

Step 5: Calculate the Fourth Term

Substitute \(n = 4\) into the general formula to find \(a_4\):\[a_4 = \frac{2^4}{2^{4+1}} = \frac{16}{2^5} = \frac{16}{32} = \frac{1}{2}\]

Key Concepts

General Term of Sequence

In mathematics, a sequence is a set of numbers in a specific order. Each number in the sequence is known as a term. To describe each member of the sequence, we often use a general formula. This formula lets us find any term in the sequence by substituting different values of "n". For the sequence given in the exercise, the general term is \(a_n = \frac{2^n}{2^{n+1}}\). Here, the formula describes how each term is related to its position number, \(n\), in the sequence.
By replacing \(n\) with any natural number, we can find that term in the sequence. This relationship holds the pattern beneath the sequence, allowing us to more deeply understand or predict other terms.

Arithmetic Calculations

Calculating terms in a sequence involves simple arithmetic operations. Starting with our general formula \(a_n = \frac{2^n}{2^{n+1}}\), we can easily find each sequence term through substitution and arithmetic steps.

• First, substitute \(n = 1\) to find the first term: \(a_1 = \frac{2^1}{2^{1+1}} = \frac{2}{4} = \frac{1}{2}\).
• Next, for the second term, substitute \(n = 2\): \(a_2 = \frac{2^2}{2^{3}} = \frac{4}{8} = \frac{1}{2}\).
• Continue this pattern for the third and fourth terms by using \(n = 3\) and \(n = 4\) respectively, calculating \(a_3 = \frac{8}{16} = \frac{1}{2}\) and \(a_4 = \frac{16}{32} = \frac{1}{2}\).

Through simple substitution and division, we uncover that each term equals \(\frac{1}{2}\). This consistent result underlines the sequence's uniformity.

Exponent Rules

Understanding exponent rules is vital when working with powers of numbers, especially in sequences. An exponent tells us how many times to multiply a number by itself. In the given formula \(\frac{2^n}{2^{n+1}}\), we frequently use these rules:

**Rule of Division of Exponents:** When dividing like bases, subtract the exponent in the denominator from the exponent in the numerator. For example, \(\frac{2^n}{2^{n+1}}\) simplifies to \(2^{n-(n+1)} = 2^{-1}\), which equals \(\frac{1}{2}\).

This rule allows us to simplify the sequence term easily, illustrating how knowing these rules can simplify complex-looking expressions into manageable calculations. By mastering exponent laws, we can tackle many algebraic problems with confidence.