Problem 24 - Algebra and Trigonometry

Question

Write a formula for the general term (the nth term) of each arithmetic sequence. Do not use a recursion formula. Then use the formula for \(a_{n}\) to find \(a_{20}\), the 20 th term of the sequence. $$ 2,7,12,17, \dots $$

Step-by-Step Solution

Answer

The 20th term of the given arithmetic sequence is 97.

Step 1: Identify the common difference

The common difference (d) of an arithmetic sequence is the difference between any two successive terms. It is constant throughout the sequence. Looking at the sequence, take the second term (7) and subtract the first term (2) to calculate the common difference: \(d = 7 - 2 = 5\).

Step 2: Write the formula for the nth term of the arithmetic sequence

The generic formula for the nth term of an arithmetic sequence is: \(a_{n} = a_{1} + (n-1)*d\), where \(a_{1}\) is the first term and d is the common difference. Substituting the known values in, the formula becomes: \(a_{n} = 2 + (n-1)*5\).

Step 3: Compute the 20th term of the sequence

Substitute n = 20 into the formula: \(a_{20} = 2 + (20-1)*5\). Simplify it to get the 20th term: \(a_{20} = 2 + 19*5 = 2 + 95 = 97\). Hence, the 20th term of the sequence is 97.

Key Concepts

Common Difference

One of the fundamental concepts in understanding arithmetic sequences is the **common difference**. In an arithmetic sequence, this value remains constant across the entire sequence and is simply the difference between successive terms.

To find the common difference, take any term in the sequence and subtract the previous term from it. For example, consider the sequence: 2, 7, 12, 17, \(\dots\). If we take the second term, 7, and subtract the first term, 2, we obtain:

• **Common Difference (\(d\))**: \(7 - 2 = 5\)

This means that each term is 5 units greater than the one before it.

When dealing with arithmetic sequences, identifying this difference correctly is vital, as it is the building block for forming the formula that can determine any term in the sequence.

Nth Term Formula

The **nth term formula** provides a method to locate any term in an arithmetic sequence, without having to list all previous terms. This formula is a crucial tool for sequence calculation and is expressed as:

\[ a_{n} = a_{1} + (n-1) \cdot d \]
Here:

• \(a_{n}\) is the nth term we want to find.
• \(a_{1}\) is the first term of the sequence.
• \(n\) is the term number we are interested in.
• \(d\) is the common difference.

Using this formula helps you jump directly to any term in the sequence efficiently. For the given sequence: 2, 7, 12, 17, \(\dots\), the general formula is:
\[ a_{n} = 2 + (n-1) \cdot 5 \]
This simplified formula results from substituting \(a_{1} = 2\) and \(d = 5\) into the general formula.

Sequence Calculation

Once you've established the common difference and the nth term formula, you can perform the **sequence calculation** to find any specific term in the series. Let's see how this plays out using our sequence example and finding the 20th term.

First, substitute n for the term number you wish to calculate—in this case, n equals 20. We already have our nth term formula:
\[ a_{n} = 2 + (n-1) \cdot 5 \]
Substitute \(n = 20\) into the formula:

• \(a_{20} = 2 + (20-1) \cdot 5\)
• Simplify: \(a_{20} = 2 + 19 \cdot 5\)
• Calculate: \(a_{20} = 2 + 95 = 97\)

Thus, the 20th term of the sequence is 97. Understanding this calculation method simplifies the process of finding terms further down the sequence without having to calculate or list every single term before it.