Problem 21 - Algebra and Trigonometry

Question

21-26 \(\approx\) Find the first five terms of the sequence, and determine whether 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. Yes, it is arithmetic; the common difference is 7.

Step 1: Identify the Given Formula

We are given the formula for the general term of the sequence: \(a_n = 4 + 7n\). Here, \(a_n\) represents the \(n\)-th term of the sequence.

Step 2: Calculate the First Five Terms of the Sequence

To find the first five terms of the sequence, substitute \(n = 1, 2, 3, 4,\) and \(5\) into the formula \(a_n = 4 + 7n\). - For \(n = 1\), \(a_1 = 4 + 7 \times 1 = 11\).- For \(n = 2\), \(a_2 = 4 + 7 \times 2 = 18\).- For \(n = 3\), \(a_3 = 4 + 7 \times 3 = 25\).- For \(n = 4\), \(a_4 = 4 + 7 \times 4 = 32\).- For \(n = 5\), \(a_5 = 4 + 7 \times 5 = 39\).Thus, the first five terms are: 11, 18, 25, 32, 39.

Step 3: Determine if the Sequence is Arithmetic

A sequence is arithmetic if there is a constant difference between consecutive terms. Calculate the differences between the terms just derived:- \(a_2 - a_1 = 18 - 11 = 7\)- \(a_3 - a_2 = 25 - 18 = 7\)- \(a_4 - a_3 = 32 - 25 = 7\)- \(a_5 - a_4 = 39 - 32 = 7\)Each difference is 7, confirming that the sequence is arithmetic with a common difference, \(d = 7\).

Step 4: Express the General Term in Standard Arithmetic Form

In an arithmetic sequence, the \(n\)-th term is given by the formula \(a_n = a + (n-1)d\), where \(a\) is the first term and \(d\) is the common difference. Here, \(a = 11\) and \(d = 7\). Substitute these values into the formula:\[ a_n = 11 + (n-1) \times 7 \]Simplifying this expression:\[ a_n = 11 + 7n - 7 = 4 + 7n \]This matches the given formula \(a_n = 4 + 7n\), confirming our results.

Key Concepts

Common Difference

In an arithmetic sequence, the **common difference** is what makes it special. It’s the constant difference between each pair of consecutive terms. This is what determines the uniformity in the sequence.
For example, let's consider our exercise:

• We calculated each term by plugging different values of \( n \) into the given general term formula.
• The terms obtained were 11, 18, 25, 32, and 39.

To confirm that this is an arithmetic sequence, we need to ensure the difference between consecutive terms is consistent:

• \( a_2 - a_1 = 18 - 11 = 7 \)
• \( a_3 - a_2 = 25 - 18 = 7 \)
• Each subsequent calculation yields the same difference of 7, verifying that 7 is the common difference \( d \).

This uniformity is what categorically makes this sequence arithmetic.

General Term Formula

The general term formula of an arithmetic sequence helps to find any term in the sequence without listing all terms.
The formula is given by: \[ a_n = a + (n-1)d \] Where:

• \( a \) is the first term of the sequence.
• \( d \) is the common difference.
• \( n \) is the position of the term in the sequence.

In our example, the general formula provided was \( a_n = 4 + 7n \).After calculating it using the arithmetic sequence formula, we validated it as follows:

• First term \( a = 11 \) (obtained from our Step 2 results).
• Common difference \( d = 7 \).
• Plugged these into the standard formula: \( a_n = 11 + (n-1)7 \).
• When simplified, it matched the given formula: \( a_n = 4 + 7n \).

This shows the power of the general term formula across mathematical sequences.

Sequence Terms

The terms of an arithmetic sequence are created by repeatedly adding the common difference starting from the first term, which sets up a predictable pattern or set of numbers.
Here's how you determine the specific terms:

• Use the general term formula to calculate the sequence.
• In our problem, you evaluate \( a_n = 4 + 7n \) with successive \( n \) values:
• When \( n = 1 \) yields the first term: \( a_1 = 11 \)
• When \( n = 2, 3, 4, 5 \) yield 18, 25, 32, and 39 respectively.

By following this process, you get all the required terms in the sequence,giving clarity in identifying where each number lies in the arithmetic pattern.

Substitution

**Substitution** is a fundamental algebraic tool, essential in producing the terms of a sequence and confirming its characteristics.
Substitution involves:

• Replacing the variable \( n \) in the general term formula with actual numbers to find particular terms in the sequence.
• For instance, by substituting \( n = 1, 2, 3, 4, 5 \) into our formula \( a_n = 4 + 7n \), we computed the first five terms of the sequence.

This method provides a concrete way to see how the sequence progresses by plugging different values into the formula.
Through substitution, we convert an abstract expression into immediate numeric values, which verifies both the arithmetic nature of the sequence and its general behavior.