Problem 14 - Algebra 2 and Trigonometry

Question

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

Step-by-Step Solution

Answer

The first five terms are 3, 2, \(\frac{5}{3}\), \(\frac{3}{2}\), and \(\frac{7}{5}\).

Step 1: Understand the Sequence Formula

The sequence is given by the formula \(a_{n} = \frac{n+2}{n}\). This means to find any term in the sequence, we need to substitute \(n\) with the desired term number.

Step 2: Calculate the First Term

To find the first term \(a_1\), substitute \(n = 1\) into the formula: \[a_1 = \frac{1+2}{1} = 3.\] So, the first term is 3.

Step 3: Calculate the Second Term

Next, find the second term \(a_2\) by substituting \(n = 2\): \[a_2 = \frac{2+2}{2} = 2.\] Thus, the second term is 2.

Step 4: Calculate the Third Term

Calculate the third term \(a_3\) by replacing \(n = 3\): \[a_3 = \frac{3+2}{3} = \frac{5}{3}.\] So, the third term is \(\frac{5}{3}\).

Step 5: Calculate the Fourth Term

Find the fourth term \(a_4\) with \(n = 4\): \[a_4 = \frac{4+2}{4} = \frac{3}{2}.\] Therefore, the fourth term is \(\frac{3}{2}\).

Step 6: Calculate the Fifth Term

Finally, calculate the fifth term \(a_5\) by substituting \(n = 5\): \[a_5 = \frac{5+2}{5} = \frac{7}{5}.\] Thus, the fifth term is \(\frac{7}{5}\).

Key Concepts

Sequence Formula

When dealing with arithmetic sequences, particularly those expressed with a formula, it is crucial to understand the structure of the sequence formula itself. The sequence we are examining here is represented by the formula \[ a_{n} = \frac{n+2}{n} \].
This formula tells us how to find any term in the sequence just by knowing the term number, represented by \( n \). In other words, the sequence formula acts as a rule or a function: you plug in a value for \( n \), and out comes the value of the sequence at that term position.

• The numerator, \( n+2 \), increases as \( n \) increases, adding a constant to \( n \).
• The denominator, \( n \), is simply the term number you are choosing to find.

This formula indicates that each term in the sequence relies entirely on the choice of \( n \), making it straightforward to explore and understand specific points in this sequence.

Term Calculation

Calculating the terms of the sequence involves substituting specific values of \( n \) into the sequence formula. For many, this sequence's pattern gradually emerges through calculating a few specific terms.
Let's break down these steps:

• For the first term \((a_1)\), substitute \( n = 1 \) into the formula: \[ a_{1} = \frac{1+2}{1} = 3. \] The result, \( 3 \), is the value of the first term.
• For the second term \((a_2)\), substitute \( n = 2 \): \[ a_{2} = \frac{2+2}{2} = 2. \] Hence, the second term is \( 2 \).

Continue this process for subsequent values of \( n \) to explore the rest of the sequence, developing an understanding of how the terms change.

Fractional Terms

Some terms in a sequence may become fractional, an essential aspect of many sequences in mathematics.
In our given sequence, from the third term onward, the terms are fractions.

• The third term \((a_3)\) calculates to: \[ a_{3} = \frac{3+2}{3} = \frac{5}{3}. \] This displays how, by substituting \( n = 3 \) into the formula, the result is a fraction.
• For the fourth term \((a_4)\), substitute \( n = 4 \): \[ a_{4} = \frac{4+2}{4} = \frac{3}{2}. \] Again, this shows a fraction.

Understanding fractional terms is vital because it highlights the variable nature of sequences as they progress. Fractions are a common result when integer operations involve division, especially when denominators differ from numerators.

Substitution Method

The substitution method is a fundamental tool in calculating sequence terms. Here, it involves replacing \( n \) in the sequence formula with specific numerical values to find the corresponding sequence term.
This method doesn't just apply to sequences but extends widely across mathematical disciplines whenever formulas are used to solve problems.

• To find the fifth term \((a_5)\), we substitute \( n = 5 \): \[ a_{5} = \frac{5+2}{5} = \frac{7}{5}. \] This substitution provides the sequence value at that position.

It's crucial to predict which value to substitute and accurately perform the arithmetic indicated by the sequence formula.By practicing substitution, solving sequence problems becomes more intuitive, and the process reinforces broader problem-solving skills in mathematics.