Problem 36 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the formula for the nth term of an arithmetic sequence

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

Step 2: Identify the known variables

From the problem, we know that the 20th term \( a_{20} = 101 \) and the common difference \( d = 3 \).

Step 3: Apply known values to the formula

Substitute the values into the formula for the 20th term: \( 101 = a_1 + (20-1) imes 3 \).

Step 4: Simplify the equation

Simplify the equation: \( 101 = a_1 + 19 imes 3 \). This becomes \( 101 = a_1 + 57 \).

Step 5: Solve for the first term \( a_1 \)

Rearrange the equation to find \( a_1 \): \( a_1 = 101 - 57 \). Thus, \( a_1 = 44 \).

Step 6: Write the formula for the nth term

Now that we have \( a_1 = 44 \) and \( d = 3 \), substitute these into the general formula to find the nth term: \( a_n = 44 + (n-1) imes 3 \).

Step 7: Simplify the formula for the nth term

Expand the expression: \( a_n = 44 + 3n - 3 \). Simplify this to \( a_n = 41 + 3n \).

Key Concepts

nth term formula

An arithmetic sequence is a list of numbers with a constant difference between consecutive terms. To find the term at a specific position, we use the **nth term formula**. This is expressed as \( a_n = a_1 + (n-1) \times d \).
Where:

• \( a_n \) is the term you want to find.
• \( a_1 \) is the first term of the sequence.
• \( n \) is the position number of the term.
• \( d \) is the common difference between terms.

To use this formula effectively, you need to know the first term and the common difference. This will allow you to calculate any term in the sequence. In our exercise, we found that the nth term formula was \( a_n = 41 + 3n \). This gives a way to calculate any term by substituting \( n \) with the term number.

common difference

The **common difference** in an arithmetic sequence is the fixed amount added to each term to get the next term. It's crucial because it defines the pattern of the sequence. This difference is denoted as \( d \) in the nth term formula.
Here's how you can find the common difference:

• Take any two consecutive terms in the sequence.
• Subtract the first term from the second term.

For example, if the sequence has a first term of 44 and a common difference of 3, then the sequence begins as 44, 47, 50, and so on. The common difference is 3 because 47 - 44 = 3.
Understanding the common difference helps determine how the sequence behaves and can assist in locating a specific term or writing the general formula for the sequence.

first term

The **first term** in an arithmetic sequence is represented by \( a_1 \) and is the starting point for the sequence. It provides a foundation from which the rest of the sequence is built. Establishing the value of the first term is essential when finding the nth term.
Once the first term is known, along with the common difference, you can use the nth term formula to establish any term in the sequence.

• For example, if the first term \( a_1 \) is 44, the sequence starts at 44. The formula then becomes \( a_n = 44 + (n-1) \times d \).
• Knowing the first term helps apply the nth term formula accurately, determining each subsequent term in the sequence.

In our given solution, we determined that the first term \( a_1 \) is 44, after solving \( 101 = a_1 + 57 \). This fetched our starting point for applying further calculations.