Problem 13 - Algebra 2 and Trigonometry

Question

In \(3-18,\) write the first five terms of each sequence. $$ a_{n}=\frac{n}{n+1} $$

Step-by-Step Solution

Answer

The first five terms are \( \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6} \).

Step 1: Introduction to Sequences

We have a sequence defined by the formula \( a_{n} = \frac{n}{n+1} \). To find the first five terms, we need to substitute the first five positive integers for \( n \) into this formula.

Step 2: Calculate the 1st Term

Substitute \( n = 1 \) into the sequence formula: \[ a_{1} = \frac{1}{1+1} = \frac{1}{2} \]. So, the first term is \( \frac{1}{2} \).

Step 3: Calculate the 2nd Term

Substitute \( n = 2 \) into the sequence formula: \[ a_{2} = \frac{2}{2+1} = \frac{2}{3} \]. So, the second term is \( \frac{2}{3} \).

Step 4: Calculate the 3rd Term

Substitute \( n = 3 \) into the sequence formula: \[ a_{3} = \frac{3}{3+1} = \frac{3}{4} \]. So, the third term is \( \frac{3}{4} \).

Step 5: Calculate the 4th Term

Substitute \( n = 4 \) into the sequence formula: \[ a_{4} = \frac{4}{4+1} = \frac{4}{5} \]. So, the fourth term is \( \frac{4}{5} \).

Step 6: Calculate the 5th Term

Substitute \( n = 5 \) into the sequence formula: \[ a_{5} = \frac{5}{5+1} = \frac{5}{6} \]. So, the fifth term is \( \frac{5}{6} \).

Step 7: Conclusion

The first five terms of the sequence are \( \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \) and \( \frac{5}{6} \).

Key Concepts

Term Calculation

When dealing with sequences in algebra, term calculation is a fundamental concept. A sequence is an ordered list of numbers, and each number is called a "term." To find each term in a sequence, we usually apply a specific formula known as the sequence formula. In our example, the sequence is given by the formula \( a_{n} = \frac{n}{n+1} \).

• To calculate the terms, you substitute values for \( n \), starting from 1, and progressing incrementally.
• For instance, to find the first term, you substitute \( n = 1 \) into the sequence formula, yielding \( a_{1} = \frac{1}{2} \).
• The subsequent terms are found by increasing \( n \): \( a_{2} = \frac{2}{3} \), \( a_{3} = \frac{3}{4} \), and so on.

This process is repeated until you have calculated the desired number of terms. Each calculation builds upon the previous one, so a solid understanding of how to manipulate these expressions is crucial.

Sequence Formula

The sequence formula is the key to unlocking the terms in a sequence. It's a mathematical expression that tells you how to calculate each term based on its position in the sequence. For our sequence, the formula is \( a_{n} = \frac{n}{n+1} \). This means the term \( a_{n} \) depends on the integer \( n \) that indicates its position.

• The numerator \( n \) is the position number itself.
• The denominator \( n+1 \) is simply one more than the position number.

This simple yet effective formula helps us determine any term in the sequence. By applying the formula, one can find not only the first few terms but continue the sequence to any point. This illustrates the power and utility of a sequence formula in mathematics.

Introduction to Sequences

Sequences are all around us, especially in mathematics. They are ordered sets of numbers that follow a particular pattern or rule. Here, we have a sequence expressed by a formula: \( a_{n} = \frac{n}{n+1} \). Understanding sequences involves recognizing patterns in numbers and seeing how these patterns are formed.

• In algebra, sequences can be thought of as a step-by-step guide to numbers that builds upon each previous step.
• Sequences can be infinite, meaning they go on forever, or finite, where they stop after a certain number of terms.

In the given example, the sequence is infinite because for every natural number \( n \), you can find a corresponding term. Learning about sequences provides a foundation for more complex mathematical concepts, making this an essential topic in any math curriculum.