Problem 40 - Algebra and Trigonometry

Question

The 20 th term of an arithmetic sequence is \(101,\) and the common difference is \(3 .\) Find a formula for the nth term.

Step-by-Step Solution

Answer

The formula for the nth term is \( a_n = 3n + 41 \).

Step 1: Understand the Problem

In this exercise, we need to find a formula for the nth term of an arithmetic sequence. We are given the 20th term of the sequence and the common difference.

Step 2: Identify the Formula for an Arithmetic Sequence

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

Step 3: Plug Known Values into the Formula

Substitute the known values into the formula: \( a_{20} = 101 \) and \( d = 3 \). So, \( 101 = a + (20-1) \times 3 \).

Step 4: Simplify the Equation

Simplify the equation by calculating \( 19 \times 3 = 57 \). This gives us \( 101 = a + 57 \).

Step 5: Solve for the First Term

Rearrange the equation to solve for \( a \), giving \( a = 101 - 57 = 44 \).

Step 6: Write the nth Term Formula

Substitute \( a = 44 \) and \( d = 3 \) back into the nth term formula to get \( a_n = 44 + (n-1)3 \).

Step 7: Simplify the Final Formula

Distribute the \( 3 \) in the formula: \( a_n = 44 + 3n - 3 \), resulting in \( a_n = 3n + 41 \).

Key Concepts

Understanding the nth Term Formula

The nth term formula is a key concept in arithmetic sequences. It helps us determine any term in the sequence using a simple equation. An arithmetic sequence is a series of numbers where each term increases or decreases by a constant value. To find the nth term of an arithmetic sequence, we use the formula:

• \( a_n = a + (n-1)\times d \),

where:

• \( a_n \) is the nth term,
• \( a \) is the first term,
• \( n \) is the term number, and
• \( d \) is the common difference.

This formula is essential because it helps us express relationships in the sequence without listing all terms. If you have a strategic approach, like solving algebraic puzzles, you'll find understanding the nth term valuable. It provides a quick way to compute any term's value by just knowing the initial conditions.

Exploring the Common Difference

The common difference in an arithmetic sequence is the consistent amount added to each term to reach the next. It's critical because it dictates the pattern of the sequence. In every arithmetic sequence, this difference remains consistent.
For example, in a sequence of 2, 5, 8, 11, the common difference is 3, because each term increases by 3. So how do we represent this mathematically?

• We denote the common difference as \( d \).
• It's calculated by subtracting any term from the term that follows it: \( d = a_{n+1} - a_n \).

Knowing the common difference is crucial because it forms part of the formula for any term in the sequence. It's what keeps the arithmetic sequence methodical and predictable. Without it, there'd be no regular pattern to follow, and predicting future terms would be impossible!

The Significance of the First Term

The first term of an arithmetic sequence is just as important as the common difference. It sets the starting point for the sequence and influences all subsequent terms. In mathematical terms, the first term is often represented by the symbol \( a \).
Consider it the anchor from which all other terms develop their values. You can think of \( a \) as a baseline, so each additional term is constructed by adding the appropriate number of common differences.

• In the formula \( a_n = a + (n-1) \times d \), the term \( a \) allows us to map out the entire sequence.
• For instance, if \( a = 2 \) and the common difference is 3, starting from \( a \), you would add 3 to get successive terms.

Understanding the first term's role helps appreciate the step-by-step buildup of terms in an arithmetic sequence. Without the first term, we can't accurately address the sequence's boundaries or calculate the precise nth term.