Problem 20 - Precalculus: Mathematical for Calculus

Question

Find the first five terms of the sequence and determine if it is arithmetic. If it is arithmetic, find the common difference and express the \(n\) th term of the sequence in the standard form $$a_{n}=a+(n-1) d$$ $$a_{n}=1+\frac{n}{2}$$

Step-by-Step Solution

Answer

The sequence is arithmetic with common difference 0.5; \(a_n = 0.5n + 1\).

Step 1: Calculate the First Term

Find the first term by substituting \(n=1\) into the given formula: \[ a_1 = 1 + \frac{1}{2} = 1.5 \]

Step 2: Calculate the Second Term

Find the second term by substituting \(n=2\) into the given formula: \[ a_2 = 1 + \frac{2}{2} = 2 \]

Step 3: Calculate the Third Term

Find the third term by substituting \(n=3\) into the given formula: \[ a_3 = 1 + \frac{3}{2} = 2.5 \]

Step 4: Calculate the Fourth Term

Find the fourth term by substituting \(n=4\) into the given formula: \[ a_4 = 1 + \frac{4}{2} = 3 \]

Step 5: Calculate the Fifth Term

Find the fifth term by substituting \(n=5\) into the given formula: \[ a_5 = 1 + \frac{5}{2} = 3.5 \]

Step 6: Determine if the Sequence is Arithmetic

To check if the sequence is arithmetic, calculate the differences between consecutive terms:\( a_2 - a_1 = 2 - 1.5 = 0.5 \)\( a_3 - a_2 = 2.5 - 2 = 0.5 \)\( a_4 - a_3 = 3 - 2.5 = 0.5 \)\( a_5 - a_4 = 3.5 - 3 = 0.5 \) Since the difference is constant, the sequence is arithmetic.

Step 7: Write the Formula for the \(n\)th Term

For an arithmetic sequence, the \(n\) th term formula is given by \[ a_n = a_1 + (n-1) d \]where \(a_1 = 1.5\) and \(d = 0.5\).Substitute these values into the formula:\[ a_n = 1.5 + (n-1) \times 0.5 = 1.5 + 0.5n - 0.5 = 0.5n + 1 \]

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is a key concept that defines the relationship between consecutive terms. Simply put, it is the amount you add (or subtract) to move from one term to the next in the sequence. To find this difference, subtraction is used between consecutive terms of the sequence.

In the exercise you are tackling, a sequence is defined by the formula \( a_n = 1 + \frac{n}{2} \). You calculated the first few terms as:

• \( a_1 = 1.5 \)
• \( a_2 = 2 \)
• \( a_3 = 2.5 \)
• \( a_4 = 3 \)
• \( a_5 = 3.5 \)

By examining the differences between these terms,:

• \( a_2 - a_1 = 2 - 1.5 = 0.5 \)
• \( a_3 - a_2 = 2.5 - 2 = 0.5 \)
• \( a_4 - a_3 = 3 - 2.5 = 0.5 \)
• \( a_5 - a_4 = 3.5 - 3 = 0.5 \)

it is clear that the common difference is \(0.5\). This constant difference confirms that the sequence is arithmetic.

Sequence Formula

The sequence formula in arithmetic sequences is used to generate terms of the sequence. It takes one known term and systematically applies the common difference to find other terms. The formula generally looks like this:

\[ a_n = a_1 + (n-1) \cdot d \]

where

• \(a_n\) is the \(n\)th term of the sequence,
• \(a_1\) is the first term,
• \(d\) is the common difference,
• and \(n\) is the term number.

In this exercise, you found the first term \(a_1\) to be 1.5, and the common difference \(d\) to be 0.5.

Replacing these values into the formula, you get:

\[ a_n = 1.5 + (n-1) \times 0.5 \]

This equation simplifies to:\[ a_n = 0.5n + 1 \]

This formula not only defines the sequence, but allows you to calculate any term in the sequence quickly without computing previous terms.

Nth Term of a Sequence

The \(n\)th term of a sequence is a way to express any term in an arithmetic sequence directly using its position or index \(n\). This feature is particularly useful for finding terms far along in the sequence without the need to list all preceding terms.

For an arithmetic sequence, the formula for the \(n\)th term is expressed as:\[ a_n = a_1 + (n-1) \cdot d \]

Through the exercise, you derived this equation for your sequence as:\[ a_n = 0.5n + 1 \]

Understanding this concept means that to find, say, the 10th term in the sequence, you just substitute \(n=10\) into this formula:

\[ a_{10} = 0.5 \times 10 + 1 = 5 + 1 = 6 \]

In a broader sense, the \(n\)th term allows you to describe the sequence algebraically, making it easier to understand the overall pattern and predict future terms.