Problem 13 - Algebra 2 and Trigonometry

Question

In \(9-14 :\) a. Find the common difference of each arithmetic sequence. b. Write the \(n\) th term of each sequence for the given value of \(n .\) $$ -1,-3,-5,-7, \ldots, n=10 $$

Step-by-Step Solution

Answer

The common difference is -2, and the 10th term is -19.

Step 1: Identify the Sequence

The given sequence is \( -1, -3, -5, -7, \ldots \). We need to find its common difference and the nth term for \( n = 10 \).

Step 2: Find the Common Difference

An arithmetic sequence has a constant difference between consecutive terms. Calculate the difference between the first two terms: \( -3 - (-1) = -3 + 1 = -2 \). Thus, the common difference \( d = -2 \).

Step 3: Write the Formula for the n-th Term

The formula for the n-th term of an arithmetic sequence is given by \( a_n = a_1 + (n-1) \cdot d \), where \( a_1 \) is the first term and \( d \) is the common difference.

Step 4: Substitute Values into the Formula

In this sequence, \( a_1 = -1 \). Substituting \( a_1 = -1 \) and \( d = -2 \) into the formula, we get: \( a_n = -1 + (n-1) \cdot (-2) \).

Step 5: Simplify the Formula

Simplify the formula: \( a_n = -1 - 2(n-1) \). Expand and simplify: \( a_n = -1 - 2n + 2 \). Thus, \( a_n = 1 - 2n \).

Step 6: Calculate the 10th Term

Substitute \( n = 10 \) into the simplified formula: \( a_{10} = 1 - 2(10) \). Calculate: \( a_{10} = 1 - 20 = -19 \).

Key Concepts

Understanding Common Difference

In an arithmetic sequence, the common difference is the consistent interval between successive terms. Identifying this constant is crucial because it defines the nature of the sequence. For instance, in the sequence \(-1, -3, -5, -7, \ldots\), our task is to find this common difference.
To determine the common difference, you simply subtract the first term from the second term in the sequence. Subtraction shows how much each term increases or decreases from the previous one. In our exercise, subtracting \(-3\) from \(-1\) gives:

• \(-3 - (-1) = -3 + 1 = -2\)

This outcome means we subtract 2 each time to get the next term. This value, \(-2\), is our common difference, indicating how quickly our sequence is changing as you move from one term to the next.

The n-th Term Formula Explained

For arithmetic sequences, the n-th term formula is a magic tool. It lets you find any term in the sequence without listing all the terms beforehand. This is particularly helpful when you're dealing with sequences that go up to very large numbers.
The n-th term formula is expressed as:

• \(a_n = a_1 + (n-1) \cdot d\)

Here, \(a_1\) is the first term of the sequence, \(d\) is the common difference, and \(n\) represents the term number you want to find. In simple terms, this formula adds the product of the common difference and one less than the term's position (\(n-1\)) to the first term.
In our sequence \(-1, -3, -5, -7, \ldots\), with \(a_1 = -1\) and \(d = -2\), we apply these values to get:

• \(a_n = -1 + (n-1) \cdot (-2)\)

This equation will produce any term number, \(n\), in our sequence!

Easy Sequence Calculation

Once you have both the common difference and the n-th term formula, calculating any specific term in the sequence becomes straightforward.
Let's say you want to find the 10th term of your sequence. You'd just plug \(n = 10\) into the n-th term formula \(a_n = 1 - 2n\).
Substitute \(n = 10\) into the formula to get:

• \(a_{10} = 1 - 2(10)\)
• \(= 1 - 20\)
• \(= -19\)

This tells us that the 10th term in our sequence is \(-19\).
With this process, you can quickly find any term in the sequence. By understanding the elements of the arithmetic sequence, performing such calculations becomes a breeze for any value of \(n\).