Problem 10 - Basic Engineering Mathematics

Question

The 6th term of an \(A P\) is 17 and the 13 th term is 38 . Determine the 19 th term.

Step-by-Step Solution

Answer

The 19th term of the AP is 56.

Step 1: Recall the formula for the nth term of an AP

The nth term of an arithmetic progression (AP) is given by the formula: \[ a_n = a + (n-1)d \] where \( a \) is the first term and \( d \) is the common difference.

Step 2: Set up equations for the given terms

We know that the 6th term is 17 and the 13th term is 38. Using the formula for the nth term, we can set up the following equations:\[ a + 5d = 17 \] \[ a + 12d = 38 \]

Step 3: Solve the system of equations

Subtract the first equation from the second to eliminate \( a \):\[ (a + 12d) - (a + 5d) = 38 - 17 \]\[ 7d = 21 \]Divide both sides by 7 to find \( d \):\[ d = 3 \]

Step 4: Substitute d back to find a

Use the value of \( d \) in one of the original equations to find \( a \). We'll use \( a + 5d = 17 \):\[ a + 5(3) = 17 \]\[ a + 15 = 17 \]\[ a = 2 \]

Step 5: Calculate the 19th term

Use the nth term formula with \( a = 2 \), \( d = 3 \) to find the 19th term:\[ a_{19} = a + 18d \]\[ a_{19} = 2 + 18(3) \]\[ a_{19} = 2 + 54 \]\[ a_{19} = 56 \]

Key Concepts

nth term formula

To determine a specific term in an arithmetic progression (AP), we rely on the nth term formula. This formula allows us to find any term in the sequence if we know the first term and the common difference. Break it down like this:

• The formula for the nth term is:
  \[a_n = a + (n-1)d\]
• Here, \(a\) is the first term of the sequence.
• \(n\) is the position of the term within the sequence.
• \(d\) represents the common difference between consecutive terms.

Using this formula, you can quickly find any term as long as the first term and the common difference are known. For instance, in the original exercise, to find the 6th term where \(a = 2\) and \(d = 3\), we substitute into the formula:
\[a_6 = 2 + (6-1) \times 3 = 17\]
Thus, with the nth term formula, arithmetic sequences become less of a mystery!

common difference

The common difference in an arithmetic progression (AP) is a key element that defines the sequence. It is the amount by which each term increases (or decreases) from the previous term. Here's the scoop on the common difference:

• Denoted by \(d\), the common difference is what makes an arithmetic sequence uniform.
• To find \(d\) when given specific terms, set up a subtraction between two consecutive terms.
• For the exercise, the common difference was found using the difference between the 6th term and the 13th term:
  \[d = \frac{38 - 17}{13 - 6} = \frac{21}{7} = 3\]

Discovering \(d\) is crucial as it applies across the entire sequence, allowing the nth term formula to be used effectively. With \(d\) known, any term in the progression can be calculated with ease.

system of equations

When determining unknowns in an arithmetic sequence, a system of equations can be an invaluable tool. This method is particularly useful when more than one unknown needs to be solved simultaneously. Here’s a simple breakdown:

• A system of equations involves two or more equations that are solved together using the same variables.
• In our exercise, the system was formed using the nth term formula for the 6th and 13th terms:
  \[a + 5d = 17\]
  \[a + 12d = 38\]
• By subtracting these two equations, we eliminate \(a\) and solve for \(d\), then use \(d\) to find \(a\).

This step-by-step process helps in finding the values of unknowns systematically. Solving the equations gives us precise values for both \(a\) and \(d\), which are essential for calculating any desired term in the sequence. It’s a straightforward technique that works well in arithmetic problems like these.