Problem 5 - Algebra 2 and Trigonometry

Question

In \(3-18,\) write the first five terms of each sequence. $$ a_{n}=2 n $$

Step-by-Step Solution

Answer

The first five terms are 2, 4, 6, 8, and 10.

Step 1: Identify the Sequence Formula

The given sequence formula is \(a_n = 2n\). This formula tells us that to find the value of any term \(a_n\), we need to multiply the term number \(n\) by 2.

Step 2: Calculate the First Term

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

Step 3: Calculate the Second Term

To find the second term \(a_2\), substitute \(n = 2\) into the formula: \(a_2 = 2 \times 2 = 4\). Hence, the second term is 4.

Step 4: Calculate the Third Term

To find the third term \(a_3\), substitute \(n = 3\) into the formula: \(a_3 = 2 \times 3 = 6\). Thus, the third term is 6.

Step 5: Calculate the Fourth Term

To find the fourth term \(a_4\), substitute \(n = 4\) into the formula: \(a_4 = 2 \times 4 = 8\). Therefore, the fourth term is 8.

Step 6: Calculate the Fifth Term

To find the fifth term \(a_5\), substitute \(n = 5\) into the formula: \(a_5 = 2 \times 5 = 10\). Thus, the fifth term is 10.

Key Concepts

Arithmetic Sequences

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is consistent. This difference is called the common difference. Arithmetic sequences can be seen in daily life, such as dates on a calendar or house numbers on a street.
An arithmetic sequence takes a straightforward pattern that follows a linear path. For example, in our sequence, each number increases by the same amount, leading to an ordered progression.
In mathematical terms, if you have a first term denoted as \(a_1\), an arithmetic sequence can be described using a formula for the nth term:

• \(a_n = a_1 + (n-1)d\)

where \(d\) is the common difference. Understanding this pattern helps tremendously in quickly determining any term in a sequence without needing to manually calculate each step.

Formula Application

Formula application in sequences is about using the given formula to find specific terms of the sequence. The process begins by identifying the general form of the sequence provided. It is crucial to know the role of \(n\), as it represents the position of the term in the sequence.
For instance, given the formula \(a_n = 2n\), you can ascertain that the formula multiplies the term number by 2. This simple arithmetic operation demonstrates how formulas can encapsulate complex patterns in a concise, easy-to-use expression.
When applying the formula, substitute the position number for \(n\) and perform the calculations. This technique affords a systematic approach to achieving consistent results, removing the need for guesswork.

Term Calculation

Performing term calculations requires substituting the sequence position into the sequence's formula. Each term in the sequence results from this substitution process.
Let's walk through the calculation of the first five terms of the sequence \(a_n = 2n\):

• The first term \(a_1 = 2 \times 1 = 2\).
• The second term \(a_2 = 2 \times 2 = 4\).
• The third term \(a_3 = 2 \times 3 = 6\).
• The fourth term \(a_4 = 2 \times 4 = 8\).
• The fifth term \(a_5 = 2 \times 5 = 10\).

By replacing \(n\) with the term number, you can see how the sequence unfolds. Each calculation reflects the rule of multiplying the position by 2, offering an efficient way to navigate any arithmetic sequence.