Problem 24 - 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 \(a_{n}=a+(n-1) d\) $$ a_{n}=1+\frac{n}{2} $$

Step-by-Step Solution

Answer

The first five terms are 1.5, 2, 2.5, 3, 3.5. It's arithmetic with d=0.5.

Step 1: Find the first five terms

\(a_1 = 1 + \frac{1}{2} = 1.5\)
\(a_2 = 1 + \frac{2}{2} = 2\)
\(a_3 = 1 + \frac{3}{2} = 2.5\)
\(a_4 = 1 + \frac{4}{2} = 3\)
\(a_5 = 1 + \frac{5}{2} = 3.5\)

Step 2: Determine if arithmetic

The common difference: \(d = a_2 - a_1 = 2 - 1.5 = 0.5\), \(a_3 - a_2 = 0.5\), etc. Since the difference is constant, this is an arithmetic sequence with common difference \(d = 0.5\).

Step 3: Express in standard form

\(a_n = a + (n-1)d = 1.5 + (n-1)(0.5) = 1.5 + 0.5n - 0.5 = 1 + 0.5n\), which matches the given formula.

Key Concepts

Sequence Terms

In any arithmetic sequence, each number is called a 'term.' These terms form the sequence, with each one following a specific order based on a formula. For instance, in the exercise, the given sequence is represented by the formula \( a_n = 1 + \frac{n}{2} \). Here, \( a_n \) is the expression that helps us calculate each term. If we plug different values of \( n \) (like 1, 2, 3…), we get the first few terms of the sequence.
To illustrate, let's find the first five terms using the given formula:

• When \( n = 1 \), the term \( a_1 = 1.5 \)
• When \( n = 2 \), the term \( a_2 = 2 \)
• When \( n = 3 \), the term \( a_3 = 2.5 \)
• When \( n = 4 \), the term \( a_4 = 3 \)
• When \( n = 5 \), the term \( a_5 = 3.5 \)

These computations show us how the formula \( a_n = 1 + \frac{n}{2} \) generates each sequence term.

Common Difference

An arithmetic sequence is characterized by the property that the difference between any two consecutive terms is constant. This constant is called the 'common difference,' denoted by \( d \). It is a crucial component of understanding arithmetic sequences because it determines the rate at which the sequence increases or decreases.
To identify if our sequence is arithmetic and find the common difference, we check the difference 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 each difference is the same, equal to 0.5, our sequence is indeed arithmetic with a common difference \( d = 0.5 \). This uniform increase confirms that the sequence is not only arithmetic but predictable in its pattern.

Nth Term Formula

The nth term of an arithmetic sequence provides the formula needed to find any term in the sequence without listing all the preceding terms. The general formula for the nth term in an arithmetic sequence is \( a_n = a_1 + (n-1)d \), where \( a_1 \) is the first term and \( d \) is the common difference.
For the sequence in our exercise, \( a_1 = 1.5 \) (as found from the first term of our sequence) and \( d = 0.5 \). Plugging these values into the formula gives us the expression for the nth term:
\[ a_n = 1.5 + (n-1) \times 0.5 \]
This formula allows us to find any term in the sequence by simply substituting the desired term number for \( n \). For example, to find the 10th term \( a_{10} \), substitute \( n = 10 \):
\[ a_{10} = 1.5 + (10-1) \times 0.5 = 1.5 + 4.5 = 6 \]
This process makes predicting large-term entries quick and straightforward, demonstrating the power of mathematical formulas in revealing pattern behaviors.