Problem 26 - Algebra and Trigonometry

Question

23-32 me the common difference, the fifth term, the \(n\)th term, and the 100th term of the arithmetic sequence. $$11,8,5,2, \dots$$

Step-by-Step Solution

Answer

Common difference: -3, Fifth term: -1, nth term: \(14 - 3n\), 100th term: -286.

Step 1: Identify the common difference

An arithmetic sequence is defined by a common difference between consecutive terms. Subtract the first term (11) from the second term (8) to find the common difference: \( 8 - 11 = -3 \). Therefore, the common difference is -3.

Step 2: Calculate the fifth term

To find the fifth term, start with the first term and add the common difference four times (since the fifth term is four steps away from the first term): \( 11 + 4(-3) = 11 - 12 = -1 \). So, the fifth term is -1.

Step 3: Derive the formula for the nth term

The formula for the \(n\)th term \(a_n\) of an arithmetic sequence is \(a_n = a_1 + (n-1)d\), where \(a_1\) is the first term and \(d\) is the common difference. Substituting \(a_1 = 11\) and \(d = -3\), we get: \[ a_n = 11 + (n-1)(-3) = 11 - 3n + 3 = 14 - 3n \].

Step 4: Calculate the 100th term

Use the \(n\)th term formula derived in the previous step. Substitute \(n = 100\) into the formula \(a_n = 14 - 3n\): \(a_{100} = 14 - 3 \times 100 = 14 - 300 = -286\). Therefore, the 100th term is -286.

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is what separates consecutive terms.
This difference remains constant as you move from one term to the next. For instance, if you're given the sequence 11, 8, 5, 2, and so on, each term is followed by the next by subtracting a fixed number, which in this case is -3.
You can find this common difference by subtracting the first term from the second term:

• Common Difference = Second Term - First Term
• Common Difference = 8 - 11 = -3

Understanding the common difference is key because it helps you predict the progression of the sequence.
It's like having a map that shows how each element relates to its neighbors. This makes calculating future terms straightforward.

nth Term Formula

With the common difference known, calculating any term in the sequence becomes a breeze with the nth term formula.
The nth term formula for an arithmetic sequence is given by:

• \( a_n = a_1 + (n-1)d \)

In this formula:

• \(a_n\) is the term you want to find.
• \(a_1\) is the first term of the sequence.
• \(n\) is the position of the term in the sequence.
• \(d\) is the common difference.

For the sequence 11, 8, 5, 2, ..., you substitute \(a_1 = 11\) and \(d = -3\):

• \[ a_n = 11 + (n-1)(-3) = 14 - 3n \]

This formula works like a magic tool, allowing you to compute any term directly, without having to list all previous terms.

Terms of a Sequence

In any arithmetic sequence, terms are the individual numbers you see.
They're connected through the common difference and positioned according to their order. Understanding terms in a sequence means recognizing each one's place and value.
With our sequence example 11, 8, 5, 2, ..., every term apart from the first is generated from the one before it by adding the common difference of -3. When you want to find a specific term, like the fifth term, you simply use the common difference:

• First term, 11, minus 3, repeatedly.
• 11, 8, 5, 2, ...
• On the fifth step: 11 + 4(-3) = -1

Each term has a story. Whether it’s the straightforward first term or the complex 100th term, understanding each term involves knowing how they relate to one another through the arithmetic process.