Problem 25 - 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 \(a_{n}=a+(n-1) d\) $$ a_{n}=6 n-10 $$

Step-by-Step Solution

Answer

The sequence is arithmetic with a common difference of 6. The nth term is \( a_n = 6n - 10 \).

Step 1: Understand the Sequence Formula

The given sequence is expressed as \( a_n = 6n - 10 \). This formula calculates each term of the sequence based on its position \( n \). To find the first five terms, we will substitute \( n \) with the numbers 1 through 5.

Step 2: Calculate the First Five Terms

Substitute \( n = 1, 2, 3, 4, 5 \) into the formula \( a_n = 6n - 10 \) to get the first five terms.\- For \( n=1 \), \( a_1 = 6(1) - 10 = -4 \).\- For \( n=2 \), \( a_2 = 6(2) - 10 = 2 \).\- For \( n=3 \), \( a_3 = 6(3) - 10 = 8 \).\- For \( n=4 \), \( a_4 = 6(4) - 10 = 14 \).\- For \( n=5 \), \( a_5 = 6(5) - 10 = 20 \). \The first five terms are -4, 2, 8, 14, and 20.

Step 3: Determine if the Sequence is Arithmetic

A sequence is arithmetic if the difference between consecutive terms is constant. Calculate the difference between each pair of consecutive terms: \- \( a_2 - a_1 = 2 - (-4) = 6 \) \- \( a_3 - a_2 = 8 - 2 = 6 \) \- \( a_4 - a_3 = 14 - 8 = 6 \) \- \( a_5 - a_4 = 20 - 14 = 6 \) \Since the common difference is consistently 6, the sequence is arithmetic.

Step 4: Express the nth Term in the Standard Arithmetic Form

The formula for the \( n \) th term of an arithmetic sequence is given by \( a_n = a + (n-1) d \), where \( a \) is the first term and \( d \) is the common difference. From our findings: \- The first term \( a = -4 \). \- The common difference \( d = 6 \). \So, the \( n \) th-term formula can be written as \( a_n = -4 + (n-1) \times 6 \). Simplifying this: \\[ a_n = -4 + 6n - 6 = 6n - 10 \] \The standard form matches the original formula, confirming our findings.

Key Concepts

Understanding the Common Difference

In an arithmetic sequence, the common difference is the key element that ensures the terms are evenly spaced. When you look at a list of numbers that form an arithmetic sequence, you will notice that you can get from one term to the next by adding the same number each time. This number is what we call the "common difference".

In the provided example, we found the sequence to be arithmetic because, after calculating the differences between consecutive terms,

• The difference between the second term and the first term, \( a_2 - a_1 \), is 6.
• This same difference appeared between the third and second terms (\( a_3 - a_2 \)), the fourth and third terms (\( a_4 - a_3 \)), and so on.

With a constant common difference of 6, this confirms that the sequence is arithmetic. If the common difference isn't constant, then the sequence isn't arithmetic.

The nth Term Formula Explained

The formula for finding any term in an arithmetic sequence is described by the expression: \[a_n = a + (n-1) \times d \]. Here, \( a \) represents the first term in the sequence, \( d \) is the common difference, \( n \) is the position of the term in the sequence, and \( a_n \) is the value of the term at position \( n \).

This formula allows you to find any term in the sequence without having to write out all preceding terms. For our example, where the given sequence formula is \( a_n = 6n - 10 \):

• The first term, \( a = -4 \).
• The common difference, \( d = 6 \).
• Substituting into the standard nth term formula \( a_n = -4 + (n-1) \times 6 \), simplifies back to \( 6n - 10 \), confirming it's consistent with the original formula.

This makes it easy to compute any desired term in the sequence.

Creating a Sequence of Terms

When constructing a sequence of terms in an arithmetic sequence, you start with the first term and repeatedly add the common difference to obtain subsequent terms. This systematic method ensures that the sequence maintains its orderly pattern.

In the original problem, we derived the first five terms from the nth term formula \( a_n = 6n - 10 \):

• For \( n=1 \), the term is \(-4\).
• For \( n=2 \), the term is \(2\).
• For \( n=3 \), the term is \(8\).
• For \( n=4 \), the term is \(14\).
• For \( n=5 \), the term is \(20\).

By repeating the same process of substituting \( n \) into \( a_n = 6n - 10 \), you can quickly build a sequence. This predictive approach is incredibly useful for identifying terms at further positions in the sequence without starting from the beginning each time.