Problem 12 - Algebra and Trigonometry

Question

Show that each sequence is arithmetic. Find the common difference, and list the first four terms. $$ \left\\{a_{n}\right\\}=\\{4-2 n\\} $$

Step-by-Step Solution

Answer

The sequence is arithmetic with a common difference of \( -2 \). The first four terms are 2, 0, -2, and -4.

Step 1: Identify the General Term

The given sequence is defined by the general term \(a_n = 4 - 2n\). This formula gives us the value of the nth term of the sequence.

Step 2: Calculate the First Four Terms

Compute the first four terms by substituting \(n = 1, 2, 3, 4\) into the general term: \a_1 = 4 - 2(1) = 2\, \a_2 = 4 - 2(2) = 0\, \a_3 = 4 - 2(3) = -2\, \a_4 = 4 - 2(4) = -4\.

Step 3: Determine the Common Difference

The common difference \(d\) of an arithmetic sequence is given by \(a_{n+1} - a_n\). Using the first four terms computed: \a_2 - a_1 = 0 - 2 = -2\, \a_3 - a_2 = -2 - 0 = -2\, \a_4 - a_3 = -4 - (-2) = -2\. Hence, the common difference is \(d = -2\).

Step 4: Verify the Arithmetic Sequence Property

Confirm that the sequence \(a_n = 4 - 2n\) is arithmetic since the common difference \(d\) is consistent (always \(-2)\).

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is a constant value that is added (or subtracted) to each term to get the next term. To find the common difference, you subtract any term from the term that follows it.

• For the sequence given by the general term \(a_n = 4 - 2n\)
• The first four terms are: 2, 0, -2, -4
• You calculate the difference between these terms:
• \(a_2 - a_1 = 0 - 2 = -2\)
• \(a_3 - a_2 = -2 - 0 = -2\)
• \(a_4 - a_3 = -4 - (-2) = -2\)

This confirms that the common difference for this sequence is \(d = -2\). This common difference is what makes the sequence arithmetic.

General Term

The general term of an arithmetic sequence is a formula that allows you to find the value of any term in the sequence. It is typically written as \(a_n\), where \(n\) indicates the position of the term in the sequence.

For the given sequence, the general term is \(a_n = 4 - 2n\). This means that to find any term in the sequence, you just substitute the term position number into \(n\). For example:

• To find the first term (\(a_1\)), substitute 1 for \(n\): \(a_1 = 4 - 2 \times 1 = 2\)
• To find the second term (\(a_2\)), substitute 2 for \(n\): \(a_2 = 4 - 2 \times 2 = 0\)
• To find the third term (\(a_3\)), and so on: \(a_3 = 4 - 2 \times 3 = -2\)

The general term is important because it provides a direct way to calculate any term in the sequence without having to know all the preceding terms.

Nth Term Calculation

Calculating the nth term in an arithmetic sequence is straightforward if you know the general term formula. For the sequence defined by \(a_n = 4 - 2n\), you can find the nth term by substituting \(n\) with the position number of the term you need.

For instance:

• To find the 5th term (\(a_5\)), substitute \(n = 5\) into the general term formula: \(a_5 = 4 - 2 \times 5 = 4 - 10 = -6\)
• If you want the 10th term (\(a_{10}\)), substitute \(n = 10\): \(a_{10} = 4 - 2 \times 10 = 4 - 20 = -16\)

This technique helps you quickly identify any term in the sequence, making it easier to identify patterns or specific values.

Sequence Verification

To verify whether a sequence is arithmetic, you need to check if the difference between consecutive terms is constant. If it's constant, the sequence is indeed arithmetic.

• Start with known terms: \(2, 0, -2, -4\)
• Calculate the differences: \(a_2 - a_1 = 0 - 2 = -2\), \(a_3 - a_2 = -2 - 0 = -2\), \(a_4 - a_3 = -4 - (-2) = -2\)

Since all the differences are equal (\(-2\)), the sequence satisfies the property of an arithmetic sequence.

This approach confirms that the pattern is consistent, and the given formula \(a_n = 4 - 2n\) indeed generates an arithmetic sequence with a common difference of \(-2\).