Problem 12 - Basic Engineering Mathematics

Question

Find the sum of the first 12 terms of the series \(5,9,13,17, \ldots\)

Step-by-Step Solution

Answer

The sum of the first 12 terms is 324.

Step 1: Identify the Sequence Type

The series given is an arithmetic sequence because the difference between any two consecutive terms is constant. Let's find this common difference, known as the common difference \(d\).

Step 2: Determine the Common Difference

Take the difference between the second term and the first term: \(9 - 5 = 4\). Therefore, the common difference \(d = 4\).

Step 3: Write the General Formula

The general formula to find the \(n\)-th term of an arithmetic sequence is \(a_n = a_1 + (n-1) \cdot d\). Here, \(a_1 = 5\) and \(d = 4\).

Step 4: Calculate the 12th Term

Use the formula to find the 12th term: \(a_{12} = 5 + (12-1) \cdot 4\). Calculate: \(a_{12} = 5 + 44 = 49\). So the 12th term is 49.

Step 5: Use the Sum Formula for Arithmetic Series

The sum \(S_n\) of the first \(n\) terms of an arithmetic series is given by \(S_n = \frac{n}{2} \cdot (a_1 + a_n)\). Here, \(n = 12\), \(a_1 = 5\), and \(a_{12} = 49\).

Step 6: Calculate the Sum of the First 12 Terms

Insert the values into the sum formula: \(S_{12} = \frac{12}{2} \cdot (5 + 49)\). Calculate: \(S_{12} = 6 \cdot 54 = 324\). So, the sum of the first 12 terms is 324.

Key Concepts

Sequence Type

In mathematics, sequences are ordered lists of numbers that often follow a specific pattern. An **arithmetic sequence** is a special kind of sequence where the difference between any two consecutive terms is always the same. This consistent difference is what defines the sequence as arithmetic.
In the sequence given in the exercise, which starts with 5, 9, 13, 17, ..., each number is part of an arithmetic sequence. The reason is simple: if you subtract any term from the term that follows it, the result will always be a constant number. Understanding that this sequence is arithmetic is key because it allows us to use specific formulas to calculate terms and sums related to the sequence.

Common Difference

The **common difference** in an arithmetic sequence is the consistent amount added to each term to get the next term. For the sequence in the exercise, we calculate this by finding the difference between the first two terms.
By subtracting the first term, 5, from the second term, 9, we discover that the common difference is 4. Simply put, every number in the sequence is 4 more than the one before it. Knowing the common difference helps us easily find the next numbers in the sequence and assists in other calculations related to the sequence.

n-th Term Formula

When working with arithmetic sequences, the **n-th term formula** is incredibly useful to find any term in the sequence without listing all previous terms. The formula is given by:

• \( a_n = a_1 + (n-1) \cdot d \)

where:

• \( a_n \) is the n-th term of the sequence,
• \( a_1 \) is the first term,
• \( n \) is the term number,
• \( d \) is the common difference.

For example, if you wanted to find the 12th term of the sequence from the exercise, you would plug the values into the formula: \[ a_{12} = 5 + (12-1) \cdot 4 \] Calculate to find that the 12th term is 49. This formula simplifies finding any specific term and is essential for efficient arithmetic calculations.

Series Sum Formula

The **series sum formula** is designed to find the sum of a specific number of terms in an arithmetic sequence. The formula we use is:

• \( S_n = \frac{n}{2} \cdot (a_1 + a_n) \)

where:

• \( S_n \) represents the sum of the first n terms,
• \( a_1 \) is the first term,
• \( a_n \) is the n-th term,
• \( n \) is the number of terms.

For the sequence in the exercise, where we calculate the sum of the first 12 terms, we substitute the known values: \[ S_{12} = \frac{12}{2} \cdot (5 + 49) \] On calculating, the sum is found to be 324. This formula is particularly useful because it provides the total of terms without needing to add each one separately.