Problem 2 - Algebra and Trigonometry

Question

Find the first four terms and the 100th term of the sequence. \(a_{n}=2 n+3\)

Step-by-Step Solution

Answer

The first four terms are 5, 7, 9, 11, and the 100th term is 203.

Step 1: Determine the Formula

The given sequence is defined by the formula \(a_{n} = 2n + 3\). This formula provides a way to calculate any term in the sequence based on its position \(n\).

Step 2: Calculate the First Term

Substitute \(n = 1\) into the formula: \[ a_{1} = 2(1) + 3 = 2 + 3 = 5. \]The first term is \(a_{1} = 5.\)

Step 3: Calculate the Second Term

Substitute \(n = 2\) into the formula: \[ a_{2} = 2(2) + 3 = 4 + 3 = 7. \]The second term is \(a_{2} = 7.\)

Step 4: Calculate the Third Term

Substitute \(n = 3\) into the formula: \[ a_{3} = 2(3) + 3 = 6 + 3 = 9. \]The third term is \(a_{3} = 9.\)

Step 5: Calculate the Fourth Term

Substitute \(n = 4\) into the formula: \[ a_{4} = 2(4) + 3 = 8 + 3 = 11. \]The fourth term is \(a_{4} = 11.\)

Step 6: Calculate the 100th Term

Substitute \(n = 100\) into the formula: \[ a_{100} = 2(100) + 3 = 200 + 3 = 203. \]The 100th term is \(a_{100} = 203.\)

Key Concepts

Sequence Formula

In arithmetic sequences, the sequence formula is the key to unlocking the pattern and behavior of the entire sequence. For this particular sequence, the formula is given as \(a_{n} = 2n + 3\). This formula is a mathematical way of expressing how each term in the sequence is found based on its position, \(n\). So, to find the first, second, or any other term, simply substitute the desired position of the term into the formula.
The sequence formula is important because:

• It provides clarity on how to determine any term in the sequence.
• Helps identify the pattern followed by the sequence.
• Makes calculations easier and more systematic.

Remember, the formula directly connects the position of a term with its value. If you know \(n\), you can compute \(a_{n}\) by substituting \(n\) into the sequence formula.

Term Calculation

Understanding term calculation is crucial in analyzing sequences. Using the sequence formula \(a_{n} = 2n + 3\), finding any term of the sequence becomes a straightforward substitution task. For instance, to get the first term, substitute \(n = 1\) into \(a_{n}\):\[a_{1} = 2(1) + 3 = 5.\] Similarly, to find the 100th term:\[ a_{100} = 2(100) + 3 = 203. \]For term calculation:

• Identify the term's position \(n\).
• Substitute \(n\) into the sequence formula.
• Perform arithmetic operations to calculate the term.

This repetitive process exemplifies the power of sequence formulas in locating both early and distant terms within an arithmetic sequence.

Linear Sequences

Linear sequences, like the one represented by \(a_{n} = 2n + 3\), are characterized by a constant difference between consecutive terms. This difference is known as the common difference, which is the coefficient of \(n\), in our case, it is 2. Linear sequences are also known as arithmetic sequences because of this quality.
Key points about linear sequences:

• The common difference remains consistent throughout.
• Each term increases or decreases by the common difference as \(n\) increases.
• The sequence formula follows a linear expression of \(n\).

Understanding linear sequences enhances your ability to identify patterns, predict future terms, and appreciate the simplicity of arithmetic progression. This is particularly useful in real-life situations where trends follow a constant rate of change.