Problem 30 - 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. $$15,12.3,9.6,6.9, \dots$$

Step-by-Step Solution

Answer

Common difference is -2.7, 5th term is 4.2, nth term is \(17.7 - 2.7n\), 100th term is -252.3.

Step 1: Identify the common difference

To find the common difference in an arithmetic sequence, subtract the first term from the second term. In this sequence, the first term is 15 and the second term is 12.3, so the common difference is calculated as follows: \( 12.3 - 15 = -2.7 \). The common difference is -2.7.

Step 2: Find the fifth term using the formula

The formula for the \(n\)th term of an arithmetic sequence is \( a_n = a_1 + (n-1) \cdot d \), where \(a_1\) is the first term, \(d\) is the common difference, and \(n\) is the term number. To find the fifth term (\(a_5\)), substitute \(n = 5\), \(a_1 = 15\), and \(d = -2.7\) into the formula: \( a_5 = 15 + (5-1) \cdot (-2.7) = 15 + 4 \cdot (-2.7) = 15 - 10.8 = 4.2 \). The fifth term is 4.2.

Step 3: Determine the general formula for the nth term

Using the same formula for the \(n\)th term, \( a_n = a_1 + (n-1) \cdot d \), plug in \(a_1 = 15\) and \(d = -2.7\). The general formula becomes \( a_n = 15 + (n-1) \cdot (-2.7) = 15 - 2.7n + 2.7 = 17.7 - 2.7n \). This formula can be used to find any term in the sequence.

Step 4: Calculate the 100th term of the sequence

Substitute \(n = 100\) into the general formula \( a_n = 17.7 - 2.7n \). Calculate as follows: \( a_{100} = 17.7 - 2.7 \cdot 100 = 17.7 - 270 = -252.3 \). Thus, the 100th term of the sequence is -252.3.

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is a key feature that defines the relationship between consecutive terms. It's simply the result you get when you subtract any term from the following one. In the sequence provided — 15, 12.3, 9.6, and so on — the first thing we do is calculate the common difference. We do this by subtracting the first term from the second: - Common Difference = 12.3 - 15 This calculation results in -2.7. This means that each term is 2.7 less than the previous one. Understanding the common difference helps us easily find other terms in the sequence.

Nth Term Formula

The nth term formula allows you to find any term in an arithmetic sequence without listing all the terms. The formula is written as:- \( a_n = a_1 + (n-1) \cdot d \)Where:

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

Using this formula, you can quickly jump to any specified term in a sequence by plugging in the values for \( a_1 \), \( d \), and \( n \). The formula is particularly useful for lengthy sequences.

Sequence Patterns

Understanding sequence patterns in arithmetic sequences involves recognizing the linear nature of the terms. Each term is evenly spaced from the next, thanks to the constant common difference. For instance, in our sequence 15, 12.3, 9.6, and so on, we observed a decrease by 2.7 per term. This consistent alteration is part of what makes arithmetic sequences predictable and easy to work with. When you know the common difference and the first term, the whole sequence can be developed using these linear steps.

Term Calculation

Calculating specific terms in an arithmetic sequence requires applications of the nth term formula. Let's break this down with a couple of examples from the sequence we're working on:1. **Fifth Term:** - Use the formula \( a_n = a_1 + (n-1) \cdot d \) - Plug in the values: \( a_1 = 15, n = 5, d = -2.7 \) - Calculation: \( a_5 = 15 + (5-1) \cdot (-2.7) = 4.2 \) - Thus, the fifth term is 4.2.2. **100th Term:** - Again, employ the formula \( a_n = 17.7 - 2.7n \) derived from the nth term formula - Replace \( n \) with 100: \( a_{100} = 17.7 - 2.7 \cdot 100 = -252.3 \) - The 100th term is -252.3.These calculations provide a glimpse into the efficiency of using the nth term formula over manual counting.