Problem 18 - Algebra and Trigonometry

Question

Find the nth term of the arithmetic sequence \(\left\\{a_{n}\right\\}\) whose first term \(a_{1}\) and common difference d are given. What is the 51st term? $$ a_{1}=-2 ; \quad d=4 $$

Step-by-Step Solution

Answer

The 51st term is 198.

Step 1: Understand the Formula

The nth term of an arithmetic sequence can be found using the formula: \[ a_n = a_1 + (n-1) \times d \]where \(a_n\) is the nth term, \(a_1\) is the first term, \(d\) is the common difference, and \(n\) is the term number.

Step 2: Identify Given Values

From the problem, the first term \(a_1\) is given as -2 and the common difference \(d\) is 4. We need to find the 51st term, so \(n = 51\).

Step 3: Substitute Values into the Formula

Substitute \(a_1 = -2\), \(d = 4\), and \(n = 51\) into the formula: \[ a_n = -2 + (51-1) \times 4 \]

Step 4: Simplify the Expression

First, compute the term inside the parentheses: \[ 51-1 = 50 \]Now, multiply this result by the common difference \(d\): \[ 50 \times 4 = 200 \]Finally, add this result to the first term: \[ a_{51} = -2 + 200 \]\[ a_{51} = 198 \]

Key Concepts

nth term

One of the core concepts in understanding arithmetic sequences is the idea of the 'nth term'. The nth term is a way to find any term in a sequence without listing all the terms. You can think of it as a shortcut.
In our example exercise, we wanted to find the 51st term of the sequence. To do this, we use the formula for the nth term of an arithmetic sequence. The formula helps us jump directly to the term we need, without calculating each intermediate term. Simply put, if you know the first term and the common difference, you can find any term in the sequence by plugging the values into the formula.

common difference

The 'common difference' is another key concept in arithmetic sequences. It’s the amount that each term in the sequence increases or decreases from the previous term.
For example, in the exercise provided, the common difference (\(d\)) is 4. This means that every term in the sequence is 4 more than the previous one.
Recognizing the common difference allows us to understand the pattern of the sequence. It's what makes the sequence 'arithmetic'.
If the common difference is positive, like in our example, the terms get bigger. If it's negative, the terms get smaller. For zero common difference, all terms are the same.

sequence formula

The 'sequence formula' is the general formula used to find the nth term of an arithmetic sequence. In our exercise, the formula is: \[ a_n = a_1 + (n-1) \times d \]
This formula tells you that to find any term (\(a_n\)), you start with the first term (\(a_1\)), then add the common difference (\(d\)) multiplied by one less than the term number you want (\(n-1\)).
Using this formula, we calculated the 51st term as follows:

• Substitute given values: \(a_1 = -2\), \(n = 51\), and \(d = 4\)
• Compute inside parentheses: \(51-1 = 50\)
• Multiply by common difference: \(50 \times 4 = 200\)
• Add to the first term: \(-2 + 200 = 198\)

Finally, we found \(a_{51} = 198 \). This method can be used for any term in an arithmetic sequence by simply adjusting the values in the formula.