Problem 9 - Algebra and Trigonometry

Question

Show that each sequence is arithmetic. Find the common difference, and list the first four terms. $$ \left\\{a_{n}\right\\}=\\{2 n-5\\} $$

Step-by-Step Solution

Answer

The common difference is \(2\) and the first four terms are \(-3, -1, 1, 3\).

Step 1: Identify the General Term

The given sequence is defined by the general term: \(a_{n} = 2n - 5\). This formula will be used to generate the terms of the sequence.

Step 2: Generate the First Four Terms

To find the first four terms, substitute \(n = 1\), \(n = 2\), \(n = 3\), and \(n = 4\) into the general term \(a_n = 2n - 5: \) For \(n = 1\): \(a_1 = 2(1) - 5 = -3\) For \(n = 2\): \(a_2 = 2(2) - 5 = -1\) For \(n = 3\): \(a_3 = 2(3) - 5 = 1\) For \(n = 4\): \(a_4 = 2(4) - 5 = 3\)

Step 3: Verify Arithmetic Sequence

To verify that the sequence is arithmetic, subtract the first term from the second term, the second term from the third term, and the third term from the fourth term: \(a_2 - a_1 = -1 - (-3) = 2\)\(a_3 - a_2 = 1 - (-1) = 2\)\(a_4 - a_3 = 3 - 1 = 2\)Since all these differences are the same, the sequence is arithmetic.

Step 4: State the Common Difference

The common difference \(d\) for the arithmetic sequence is \(2\).

Step 5: List the First Four Terms Again

The first four terms of the sequence are \(-3, -1, 1, 3\).

Key Concepts

common difference

In an arithmetic sequence, the common difference is a key element. It’s the constant amount that you add to each term to get to the next term. For example, in the sequence given by the general term \(a_{n} = 2n-5\), you find the common difference by subtracting one term from the next.

• Step 1: Identify two successive terms, say \(a_2\) and \(a_1\).
• Step 2: Subtract the first term from the second term: \(d = a_2 - a_1\).

In our example, \(a_2 = -1\) and \(a_1 = -3\). Therefore, the common difference \(d = -1 - (-3) = 2\). Notice that the common difference is the same no matter which consecutive terms you choose.

general term

The general term of an arithmetic sequence helps to determine any term in the sequence. It's usually written in the form \(a_{n} = a_1 + (n-1)d\), where \(a_1\) is the first term and \(d\) is the common difference. For our sequence, \(a_{n} = 2n - 5\), the structure of the formula already shows how each term is calculated: \(2n\) indicates the arithmetic component, and \(-5\) is the initial adjustment to get the specific terms for this sequence.

sequence verification

To verify if a sequence is arithmetic, ensure the difference between successive terms is constant. Here's how:

• First, calculate the first few terms using the general term formula.
• Next, find the differences between these terms.

If all these differences are equal, the sequence is arithmetic. For instance, our sequence with \(a_n = 2n - 5\) has first four terms \(-3, -1, 1, 3\). The common difference is consistently \(2\), confirming it's an arithmetic sequence.

first four terms

The first four terms of a sequence are crucial for identifying its pattern and verifying its properties. To find these terms from the general term \(a_n = 2n - 5\):

• For \(n = 1\): \(a_1 = 2(1) - 5 = -3\).
• For \(n = 2\): \(a_2 = 2(2) - 5 = -1\).
• For \(n = 3\): \(a_3 = 2(3) - 5 = 1\).
• For \(n = 4\): \(a_4 = 2(4) - 5 = 3\).

Thus, the first four terms are \(-3, -1, 1, 3\). These steps cement understanding of how each term is derived and aid in recognizing the sequence as arithmetic.