Problem 27 - Precalculus: Mathematics for Calculus

Question

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. The sequence is arithmetic with a common difference of 7, and \( a_n = 4 + 7n \).

Step 1: Identify the Given Formula

The given expression for the sequence of terms is \( a_n = 4 + 7n \). This is a general formula that will help generate each term in the sequence.

Step 2: Calculate the First Five Terms

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

Step 3: Determine if the Sequence is Arithmetic

A sequence is arithmetic if the difference between consecutive terms is constant. Calculate the difference:- \( a_2 - a_1 = 18 - 11 = 7 \)- \( a_3 - a_2 = 25 - 18 = 7 \)- \( a_4 - a_3 = 32 - 25 = 7 \)- \( a_5 - a_4 = 39 - 32 = 7 \)Since the difference is constant (7), the sequence is arithmetic.

Step 4: Express the Sequence in Standard Form

For an arithmetic sequence, the nth term is expressed as \( a_n = a + (n-1)d \), where \( a \) is the first term, and \( d \) is the common difference.Here, \( a = 11 \) and \( d = 7 \). Therefore, the expression becomes:\[ a_n = 11 + (n-1)7 \]Simplifying further, \( a_n = 11 + 7n - 7 = 4 + 7n \), confirming the given sequence matches this arithmetic form.

Key Concepts

Common Difference

A key feature of an arithmetic sequence is something called the 'common difference'. This is the difference between any two consecutive terms within the sequence. For a sequence to be arithmetic, this common difference must remain the same between each pair of consecutive terms.

To find the common difference, simply subtract the first term from the second term, the second term from the third term, and so on. For example, if you have an arithmetic sequence like our exercise, you can find the common difference by performing the following calculations:

• Subtract the first term from the second term: \(18 - 11 = 7\)
• Subtract the second term from the third: \(25 - 18 = 7\)
• Continue this for all term pairs: \(32 - 25 = 7\) and \(39 - 32 = 7\).

Notice how each of these differences is 7. Therefore, our common difference \(d\) is 7. This constancy is what qualifies our sequence as arithmetic.

General Term Formula

In arithmetic sequences, one of the most useful tools is the general term formula. This formula is written as: \(a_{n} = a + (n-1)\times d\). Here, \(a\) represents the first term of the sequence, \(d\) is the common difference, and \(n\) corresponds to the term number.

Using the general term formula allows you to find any term in the sequence without having to list all the previous terms. Let's consider our sequence from the exercise:

• First term \(a = 11\)
• Common difference \(d = 7\)

Plug these values into the formula:
\[a_{n} = 11 + (n - 1) \times 7\].When you distribute and simplify, it confirms the sequence as \(a_n = 11 + 7n - 7 = 4 + 7n \). This expression matches the general term given in the exercise, showing you the power of this formula in expressing arithmetic sequences.

Sequence Identification

Identifying whether a given sequence is arithmetic involves verifying that it meets specific characteristics. The primary trait to confirm is whether the sequence has a constant common difference between each pair of consecutive terms.

Given the sequence from the problem as \(a_n = 4 + 7n\), you can deduce that it might be arithmetic by examining the form derived earlier (\(a_n = 11 + (n-1)\times7\)) and find that we have an arithmetic sequence on our hands.

Beyond calculations, you can easily check the following:

• Evaluate several terms using the formula.
• Ensure each step reflects the common difference observed beforehand.

In our example, by generating the terms \(11, 18, 25, 32, 39\) and verifying each difference as 7, the sequence identifies clearly as arithmetic. Through this process, you improve your understanding and ability to spot arithmetic structures in a series of numbers.