Problem 38 - Algebra and Trigonometry

Question

The 12 th term of an arithmetic sequence is \(32,\) and the fifh term is 18 . Find the 20 \(\mathrm{th}\) term.

Step-by-Step Solution

Answer

The 20th term is 48.

Step 1: Establish the general formula for the arithmetic sequence

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

Step 2: Write the equations for known terms

We know the 12th term is 32, so by the formula, \( a + 11d = 32 \). We also know the 5th term is 18, so \( a + 4d = 18 \). We now have the system of equations:1. \( a + 11d = 32 \)2. \( a + 4d = 18 \)

Step 3: Solve the system of equations

Subtract the second equation from the first:\((a + 11d) - (a + 4d) = 32 - 18\)This simplifies to \(7d = 14\). Solve for \(d\) to find \(d = 2\).

Step 4: Find the first term using the common difference

Substitute \(d = 2\) into the equation \(a + 4d = 18\):\(a + 4(2) = 18\)\(a + 8 = 18\)\(a = 10\).The first term \(a\) is 10.

Step 5: Find the 20th term

Use the general formula for the 20th term:\(a_{20} = a + (20 - 1)d\)\(a_{20} = 10 + 19(2)\)\(a_{20} = 10 + 38 = 48\).Therefore, the 20th term is 48.

Key Concepts

Common Difference

In any arithmetic sequence, the common difference is the consistent amount by which each term increases (or decreases). It is denoted by the letter \( d \). This difference is what defines the sequence as 'arithmetic'. When you know two or more terms of the sequence, you can calculate this difference. For example, if the fifth term is 18 and the 12th term is 32, you can use these values to find \( d \). By subtracting the value of the earlier term from the later one and then dividing by the number of terms between them:

• Subtract: \( 32 - 18 = 14 \)
• Count the terms in between: 12th term and 5th term have 7 steps between them
• Divide the difference by the number of steps: \( 14 / 7 = 2 \)

This tells us the common difference \( d \) is 2. Understanding this concept is crucial because the entire sequence's progression relies on it.

N-th Term Formula

To find any particular term in an arithmetic sequence, the \( n \)-th term formula is your go-to tool. The formula is represented as \( a_n = a + (n-1)d \), where:

• \( a_n \) is the term you want to find.
• \( a \) is the first term of the sequence.
• \( d \) represents the common difference.
• \( n \) is the position of the term in the sequence.

For instance, to find the 20th term in a sequence where the first term \( a \) is 10 and the common difference \( d \) is 2, substitute into the formula:\[a_{20} = 10 + (20-1) imes 2\]This simplifies to:\[a_{20} = 10 + 19 imes 2 = 10 + 38 = 48\]Thus, the 20th term is 48. The formula shows how each term is a simple function of its position, the first term, and the common difference. Mastering this formula allows for quick determination of any term.

System of Equations

A system of equations is a set of two or more equations that you deal with together at the same time. In the context of arithmetic sequences, you often use them to find both the first term and the common difference when given specific terms of the sequence. In our example, the 12th term is 32, and the 5th term is 18, leading us to these two equations:

• \( a + 11d = 32 \)
• \( a + 4d = 18 \)

To solve this system, subtract the second equation from the first to find the difference:\[(a + 11d) - (a + 4d) = 32 - 18\]This reduces to:\[7d = 14\]Solving for \( d \) gives \( d = 2 \). With \( d \) known, substitute it back into one of the original equations to solve for \( a \) (the first term):\[a + 4(2) = 18 \Rightarrow a + 8 = 18 \Rightarrow a = 10\]By understanding and applying a system of equations, you can unravel the properties of the sequence, specifically identifying critical elements like the first term and the common difference.