Problem 17 - 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 $$a_{n}=a+(n-1) d$$ $$a_{n}=4+7 n$$

Step-by-Step Solution

Answer

The first five terms are 11, 18, 25, 32, 39; it is arithmetic with a common difference of 7.

Step 1: Identify the formula

The sequence is given by the formula \( a_n = 4 + 7n \). We will use this formula to find the first five terms of the sequence by substituting values for \( n \).

Step 2: Find the first term (\(a_1\))

Substitute \( n = 1 \) into the formula:\[ a_1 = 4 + 7 \times 1 = 4 + 7 = 11 \]

Step 3: Find the second term (\(a_2\))

Substitute \( n = 2 \) into the formula:\[ a_2 = 4 + 7 \times 2 = 4 + 14 = 18 \]

Step 4: Find the third term (\(a_3\))

Substitute \( n = 3 \) into the formula:\[ a_3 = 4 + 7 \times 3 = 4 + 21 = 25 \]

Step 5: Find the fourth term (\(a_4\))

Substitute \( n = 4 \) into the formula:\[ a_4 = 4 + 7 \times 4 = 4 + 28 = 32 \]

Step 6: Find the fifth term (\(a_5\))

Substitute \( n = 5 \) into the formula:\[ a_5 = 4 + 7 \times 5 = 4 + 35 = 39 \]

Step 7: Check for common differences

Calculate the difference between consecutive terms to see if it is constant:- \( a_2 - a_1 = 18 - 11 = 7 \)- \( a_3 - a_2 = 25 - 18 = 7 \)- \( a_4 - a_3 = 32 - 25 = 7 \)- \( a_5 - a_4 = 39 - 32 = 7 \)Since the differences are all 7, the sequence is arithmetic with a common difference \( d = 7 \).

Step 8: Express the nth term in standard form

The standard form of an arithmetic sequence is \( a_n = a + (n-1)d \). We know \( a = 11 \) and \( d = 7 \), so:\[ a_n = 11 + (n-1) \times 7 = 11 + 7n - 7 = 4 + 7n \] This confirms the given formula is already in the standard form.

Key Concepts

Common Difference

The common difference is a fundamental element in arithmetic sequences. Knowing it helps you determine if a sequence is arithmetic. An arithmetic sequence is a series of numbers where each term after the first is the sum of the previous term plus a constant value, known as the common difference. For instance, if you look at the sequence given by the formula \( a_n = 4 + 7n \), which gives terms like 11, 18, 25, and so on, the common difference can be determined by subtracting consecutive terms.

To find this difference, subtract the first term from the second, the second from the third, and so on:

• \( a_2 - a_1 = 18 - 11 = 7 \)
• \( a_3 - a_2 = 25 - 18 = 7 \)
• \( a_4 - a_3 = 32 - 25 = 7 \)

Since the difference is consistently 7, we confirm the sequence is arithmetic, with a common difference \( d = 7 \). Understanding the common difference helps predict any term in the sequence easily.

Nth-Term Formula

The nth-term formula is the key to unlocking any term in an arithmetic sequence. It provides a way to find the value of any term without listing all terms before it. In the standard form, the formula is expressed as \( a_n = a + (n-1)d \). This formula involves three parts:

• \( a \): the first term of the sequence.
• \( n \): the term number you want to find.
• \( d \): the common difference between the terms.

Understanding this formula allows us to rewrite the given sequence formula \( a_n = 4 + 7n \) as \( a_n = 11 + (n-1) \times 7 \).

This formula helps you calculate any term without needing to compute all previous terms, saving time and avoiding errors. It's a handy tool for proper number sequencing and prediction.

Sequence Terms

In arithmetic sequences, sequence terms are the individual numbers in the series that follow a specific pattern dictated by the nth-term formula. For the sequence given \( a_n = 4 + 7n \), sequence terms are generated by substituting different values of \( n \) into the formula.

• The first term is found by setting \( n = 1 \), resulting in \( a_1 = 4 + 7 \times 1 = 11 \).
• The second term is \( a_2 = 4 + 7 \times 2 = 18 \).
• The third term is \( a_3 = 4 + 7 \times 3 = 25 \).
• The fourth term is \( a_4 = 4 + 7 \times 4 = 32 \).
• The fifth term is \( a_5 = 4 + 7 \times 5 = 39 \).

These terms reveal the increasing nature of the series by a consistent step, which is our common difference. By finding the first few terms, you introduce the pattern, making it easier to understand how the arithmetic sequence unfolds. Recognizing sequence terms aids in visualizing the entire sequence and verifying correctness.