Problem 9 - Basic Engineering Mathematics

Question

Determine (a) the ninth, and (b) the sixteenth term of the series \(2,7,12,17, \ldots .\)

Step-by-Step Solution

Answer

The ninth term is 42, and the sixteenth term is 77.

Step 1: Identify the Series Type

First, observe the given series: 2, 7, 12, 17,... It consists of numbers that appear to increase by a constant value. This indicates that it is an arithmetic series.

Step 2: Determine the Common Difference

In an arithmetic series, each term increases by the common difference. Calculate the common difference, denoted by \(d\), by subtracting the first term from the second term. Thus, \(d = 7 - 2 = 5\).

Step 3: Use the General Term Formula

The general formula for the \(n\)-th term of an arithmetic series is given by \(a_n = a_1 + (n-1) \, d\), where \(a_1\) is the first term and \(d\) is the common difference.

Step 4: Calculate the Ninth Term

Substitute \(n = 9\), \(a_1 = 2\), and \(d = 5\) into the formula: \[ a_9 = 2 + (9-1) \times 5 = 2 + 8 \times 5 = 2 + 40 = 42. \]Hence, the ninth term is 42.

Step 5: Calculate the Sixteenth Term

Substitute \(n = 16\), \(a_1 = 2\), and \(d = 5\) into the formula: \[ a_{16} = 2 + (16-1) \times 5 = 2 + 15 \times 5 = 2 + 75 = 77. \]Thus, the sixteenth term is 77.

Key Concepts

Common Difference

In an arithmetic series, the magic ingredient that determines the pattern of the sequence is called the "common difference". The common difference is the consistent amount that each term increases or decreases by. To find this value, simply subtract the first term from the second term. For example, in the sequence 2, 7, 12, 17,..., our calculation looks like this:

• Second Term - First Term = 7 - 2
• This means our common difference, or \( d = 5 \)

This value of 5 is what you will repeatedly add to get from one term to the next.
Understanding the common difference is crucial as it helps in crafting the pattern of the sequence and is a key part of calculating any future term in the series.

General Term Formula

The general term formula of an arithmetic sequence is the tool that allows you to find any term in the sequence without listing all the terms. It provides an efficient shortcut. The formula is expressed as:

• \( a_n = a_1 + (n-1) \cdot d \)

Here, \( a_n \) represents the \( n \)-th term you are trying to find.
\( a_1 \) is the first term of the sequence, and \( d \) is the common difference between consecutive terms.
Using this formula means you can jump to your term of interest without computing all the previous terms.

n-th Term Calculation

Once you have the common difference and the general term formula, calculating the \( n \)-th term is just a matter of substitution.
For example, if you are given a series like 2, 7, 12, 17,... and need to calculate the 9th term:

• Use the formula \( a_n = a_1 + (n-1) \cdot d \)
• Substitute \( a_1 = 2 \), \( d = 5 \), and \( n = 9 \)
• This gives: \( a_9 = 2 + (9-1) \cdot 5 \)
• Simplify the expression: \( a_9 = 2 + 8 \cdot 5 = 42 \)

This method can be applied to find any term, as demonstrated by also finding the 16th term: replace \( n \) with 16 to get \( a_{16} = 2 + 15 \cdot 5 = 77 \).
Remember, this process is all about clear substitution and simple arithmetic.