Problem 13 - Basic Engineering Mathematics

Question

Find the sum of the first 21 terms of the series \(3.5,4.1,4.7,5.3, \ldots\)

Step-by-Step Solution

Answer

The sum is 199.5.

Step 1: Identify the sequence type

Examine the given terms: \(3.5, 4.1, 4.7, 5.3, \ldots\) to determine if they form an arithmetic or geometric sequence. By subtracting consecutive terms, \(4.1 - 3.5 = 0.6\), \(4.7 - 4.1 = 0.6\), we confirm the sequence is arithmetic with a common difference \(d = 0.6\).

Step 2: Use the formula for the sum of an arithmetic sequence

The formula for the sum of the first \(n\) terms of an arithmetic sequence is: \[ S_n = \frac{n}{2} \times (2a + (n-1)d) \]where \( a \) is the first term, \( d \) is the common difference, and \( n \) is the number of terms. Here, \( a = 3.5 \), \( d = 0.6 \), and \( n = 21 \).

Step 3: Plug values into the formula

Substitute the values into the formula:\[ S_{21} = \frac{21}{2} \times (2 \times 3.5 + 20 \times 0.6) \]

Step 4: Calculate the expression

First, calculate the expression inside the parentheses: \(2 \times 3.5 = 7\)\(20 \times 0.6 = 12\)Adding these gives \(7 + 12 = 19\).Now calculate the full expression: \[ S_{21} = \frac{21}{2} \times 19 = 10.5 \times 19 \].

Step 5: Compute the final answer

Multiply to find \( S_{21} \):\(10.5 \times 19 = 199.5\).Thus, the sum of the first 21 terms of the sequence is \( S_{21} = 199.5 \).

Key Concepts

Sum of an Arithmetic Sequence

Understanding the sum of an arithmetic sequence is crucial when dealing with series of numbers that increase or decrease by a constant amount, called the common difference. To find the sum of an arithmetic sequence, you can use a handy formula. This formula helps you to add up all the terms without doing it manually, which can save tons of time for long sequences. The formula is:\[ S_n = \frac{n}{2} \times (2a + (n-1)d) \] where:

• \( S_n \) is the sum of the first \( n \) terms.
• \( a \) is the first term of the sequence.
• \( d \) is the common difference.
• \( n \) is the number of terms.

Start by multiplying the number of terms \( n \) by half (which is dividing by 2). Then, multiply the first term by 2 and add it to the product of \((n-1)\) and the common difference. Bring it all together by multiplying the two results. This gets you the sum of the sequence readily.

Common Difference

The common difference is a key element in arithmetic sequences. It's the consistent number you add to each term to get the next one. In essence, it tells you how the sequence grows or shrinks. To find it, simply subtract the first term from the second term. For example, in the sequence \(3.5, 4.1, 4.7, 5.3, \ldots\), the common difference is \( 0.6 \), because \( 4.1 - 3.5 = 0.6 \).Characteristics of the Common Difference:

• **Uniformity**: It stays the same throughout the sequence, which is why we call the sequence arithmetic.
• **Sign**: The common difference can be positive, negative, or even zero.
• **Simple Calculation**: Just subtract any term from the one after it to find it: \( d = a_{n+1} - a_n \).

Understanding the common difference helps in identifying sequences and in calculating the sum efficiently.

Number of Terms in a Sequence

The number of terms in a sequence, often denoted \( n \), represents how many numbers you add up in your sequence. Knowing this is essential because it directly impacts calculations involving the sequence, like finding the total sum.When given a sequence like \(3.5, 4.1, 4.7, \ldots\), you may need to determine the number of terms that meet specific criteria. Sometimes this number is provided, as in our original problem which mentioned 21 terms. In other cases, you might need to calculate it based on additional information, such as the last term in the sequence and the common difference.

• **Formula Association**: The number of terms is used in the sum formula \( S_n = \frac{n}{2} \times (2a + (n-1)d) \).
• **Growth Understanding**: It reflects how long the sequence extends, showing how far the pattern goes.
• **Calculation Base**: If needed, the number of terms can be calculated using: \( n = \frac{\text{last term} - a}{d} + 1 \).

Grasping this concept ensures that you can correctly find and verify the sum of an arithmetic sequence or search the sequence till a specific term.