Problem 3 - Algebra and Trigonometry

Question

\(3-12\) . Find the first four terms and the 100 th term of the sequence. $$ a_{n}=n+1 $$

Step-by-Step Solution

Answer

The first four terms are 2, 3, 4, and 5; the 100th term is 101.

Step 1: Identify the Sequence Formula

We are given the formula for the sequence: \( a_n = n + 1 \). This formula means that to find the \( n \)-th term of the sequence (\( a_n \)), we take \( n \) and add 1 to it.

Step 2: Find the First Term

To find the first term, substitute \( n = 1 \) into the sequence formula: \( a_1 = 1 + 1 = 2 \). So, the first term is 2.

Step 3: Find the Second Term

For the second term, substitute \( n = 2 \) into the formula: \( a_2 = 2 + 1 = 3 \). Thus, the second term is 3.

Step 4: Find the Third Term

To find the third term, use \( n = 3 \): \( a_3 = 3 + 1 = 4 \). Therefore, the third term is 4.

Step 5: Find the Fourth Term

For the fourth term, substitute \( n = 4 \) into the sequence formula: \( a_4 = 4 + 1 = 5 \). The fourth term is 5.

Step 6: Calculate the 100th Term

To find the 100th term of the sequence, use \( n = 100 \): \( a_{100} = 100 + 1 = 101 \). Thus, the 100th term is 101.

Key Concepts

Sequence Formula

Understanding the sequence formula is crucial to solving problems about sequences. A sequence is simply a set of numbers that follow a specific pattern. For arithmetic sequences, the pattern is determined by a formula.
In this exercise, you are given the formula \( a_n = n + 1 \). This represents an arithmetic sequence where each term can be calculated by plugging values of \( n \) into the formula.
This formula is quite straightforward:

• \( n \) is the position number of the term you want to find within the sequence.
• The expression \( n + 1 \) indicates adding 1 to your position number \( n \) to calculate the value of a term.

The beauty of this sequence formula is that it provides a simple linear equation, making it easier to calculate any term without having to list out all previous terms. Learning how to interpret and use the sequence formula effectively is foundational in working with sequences in mathematics.

Term Calculation

To understand how each term in a sequence is calculated, we focus on substituting specific values of \( n \) into the sequence formula.
Our given sequence formula is \( a_n = n + 1 \), which simplifies the process of finding terms because it doesn't involve complex operations.
When calculating the nth term:

• Insert the desired term position number \( n \) into the formula.
• Compute the result by carrying out the arithmetic operation given, in this case, adding 1 to the value of \( n \).

Let's look at a few examples from the exercise:

• For the first term, \( n = 1 \): \( a_1 = 1 + 1 = 2 \).
• For the second term, \( n = 2 \): \( a_2 = 2 + 1 = 3 \).
• Similarly, for the fourth term, \( n = 4 \): \( a_4 = 4 + 1 = 5 \).

Term calculation becomes systematic and consistent with practice. Mastering this process allows you to quickly determine specific terms without iteratively listing all preceding terms.

Finding nth Term

Finding the nth term of a sequence is all about using the sequence formula effectively. This approach offers a quick path to determine any term in a sequence, especially when dealing with very large term positions.
By using the formula \( a_n = n + 1 \), you can leap to any term directly. This is useful when you need terms far along in the sequence, such as the 100th term.
To find the 100th term:

• Identify that the sequence formula is \( a_n = n + 1 \).
• Substitute \( n = 100 \) into the formula.
• Calculate \( a_{100} = 100 + 1 = 101 \).

This method saves time and effort. You aren't required to list the entire sequence up until that term. The arithmetic approach to finding the nth term also solidifies your understanding of how sequences function. It showcases the power of mathematical formulas by turning a potentially tedious task into a straightforward exercise. The more you practice these calculations, the more intuitive they become.