Problem 11 - Basic Engineering Mathematics

Question

Determine the number of the term whose value is 22 in the series \(2 \frac{1}{2}, 4,5 \frac{1}{2}, 7, \ldots\).

Step-by-Step Solution

Answer

The term number with value 22 is the 14th term.

Step 1: Identify the Pattern

First, let's look at the given series: \(2 \frac{1}{2}, 4, 5 \frac{1}{2}, 7, \ldots\). Notice that the series increases by a constant amount each time. The first term is \(2.5\), the second is \(4\), the third is \(5.5\), and the fourth is \(7\). Subtract the first term from the second term: \(4 - 2.5 = 1.5\). Similarly, subtract the second term from the third to confirm the pattern: \(5.5 - 4 = 1.5\). The series increases by \(1.5\) consistently.

Step 2: Formulate the General Term

The general term for an arithmetic series is given by \(a_n = a_1 + (n-1) \cdot d\), where \(a_1\) is the first term, \(d\) is the common difference, and \(n\) is the term number. In this series, \(a_1 = 2.5\) and \(d = 1.5\). Hence, \(a_n = 2.5 + (n-1) \cdot 1.5\).

Step 3: Set Up the Equation to Find the Term Number

We need to find the term \(a_n\) that equals 22. Plug 22 into the equation: \(22 = 2.5 + (n-1) \cdot 1.5\).

Step 4: Solve for the Term Number

Simplify the equation: 1. Subtract 2.5 from both sides: \(22 - 2.5 = (n-1) \cdot 1.5\). This gives us \(19.5 = (n-1) \cdot 1.5\).2. Divide both sides by 1.5: \(\frac{19.5}{1.5} = n-1\). Simplifying, we get \(13 = n-1\).3. Add 1 to both sides: \(n = 13 + 1 = 14\).

Step 5: Conclusion

The term number that has the value 22 in the series is the 14th term.

Key Concepts

General Term Formula

The general term formula is an essential tool in understanding arithmetic sequences. It allows you to find any term within the sequence using a simple formula rather than calculating each term sequentially. The formula for the general term of an arithmetic sequence is given by \( a_n = a_1 + (n-1) \cdot d \). Here, \( a_n \) is the term you wish to find, \( a_1 \) is the first term of the sequence, \( n \) is the term number, and \( d \) is the common difference between terms.
To utilize this formula:

• Identify the first term \( a_1 \).
• Determine the common difference \( d \).
• Substitute \( n \), \( a_1 \), and \( d \) into the formula to find \( a_n \).

This formula is particularly useful if you know the starting point and the pattern of the sequence, as it allows you to calculate any distant term without listing all previous terms.

Common Difference

The common difference \( d \) is a crucial facet of an arithmetic sequence, highlighting the consistent increment between consecutive terms. To find the common difference, simply subtract any term from the subsequent term. For instance, in the sequence given, \(4 - 2.5 = 1.5\) and \(5.5 - 4 = 1.5\) both yield the common difference of \( 1.5 \).
The importance of the common difference includes:

• Ensuring the sequence follows an arithmetic pattern.
• Formulating the general term formula accurately.
• Predicting future terms in the sequence.

Identifying the common difference confirms the pattern of change in the sequence is uniform, allowing for simplified calculations and ensuring the arithmetic nature of the sequence.

Sequence Patterns

Recognizing sequence patterns is vital to understanding the structure of an arithmetic series. Sequence patterns are what make an arithmetic sequence predictable and straightforward to work with. In an arithmetic sequence, the pattern is a regular increase or decrease between the terms, defined by the common difference \( d \).
To identify a sequence pattern, follow these steps:

• Observe a few initial terms.
• Calculate differences between consecutive terms to check for consistency.
• Use these differences to predict subsequent terms.

Understanding these patterns allows you to apply the general term formula efficiently and predict any term within the sequence without laying out each term one by one. Recognizing and confirming these patterns ensures that calculations and predictions made about the sequence are precise and reliable.