Problem 6 - Algebra 2 and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the Sequence Formula

The formula given for the sequence is \( a_n = \frac{1}{n} \). This means that for any term \( n \), the value of the term in the sequence is the reciprocal of \( n \). In simpler terms, to find the \( n^{th} \) term, you divide 1 by \( n \).

Step 2: Calculate the First Term

To find the first term of the sequence, substitute \( n = 1 \) into the formula: \( a_1 = \frac{1}{1} = 1 \). So, the first term is 1.

Step 3: Calculate the Second Term

For the second term, use \( n = 2 \) in the formula: \( a_2 = \frac{1}{2} \). Thus, the second term is \( \frac{1}{2} \).

Step 4: Calculate the Third Term

Replace \( n = 3 \) into the formula to get the third term: \( a_3 = \frac{1}{3} \). Therefore, the third term is \( \frac{1}{3} \).

Step 5: Calculate the Fourth Term

Use \( n = 4 \) to find the fourth term: \( a_4 = \frac{1}{4} \). So the fourth term is \( \frac{1}{4} \).

Step 6: Calculate the Fifth Term

Finally, to find the fifth term, substitute \( n = 5 \) into the formula: \( a_5 = \frac{1}{5} \). Thus, the fifth term is \( \frac{1}{5} \).

Key Concepts

Sequence Formula

A sequence formula is like a rulebook for sequences. Think of it as a blueprint that informs us how to build each term in a sequence. In arithmetic sequences, this formula helps us determine how each term relates to its position in the order. In the given exercise, the sequence is described by the formula: \( a_n = \frac{1}{n} \). This is known as the reciprocal sequence.

• The letter \( a \) represents the term we want to find.
• The subscript \( n \) denotes the position of the term—this is what changes as we move through the sequence.
• The formula \( \frac{1}{n} \) indicates that each term is the reciprocal of its position.

The beauty of the sequence formula in our example is its simplicity. Plug any value of \( n \) into the formula, and it spits out the specific term. So, the formula is not just a starting point; it’s our map through the entire sequence!

Term Calculation

Calculating terms in a sequence involves substituting values into your sequence formula. Let's break down how you go about this process:Start with identifying the value of \( n \). The value of \( n \) (1, 2, 3, and so forth) determines which term you are trying to find. Each value of \( n \) corresponds to a step in the sequence.Once you've chosen \( n \), plug it into the formula. In this sequence, substitute \( n \) into \( a_n = \frac{1}{n} \) and carry out the division.

• For \( n = 1 \), \( a_1 = \frac{1}{1} = 1 \)
• For \( n = 2 \), \( a_2 = \frac{1}{2} \)
• For \( n = 3 \), \( a_3 = \frac{1}{3} \)

By understanding and applying this method, you can calculate any term in the sequence quickly and effectively. Once you master this, arithmetic sequences and their calculations appear far less daunting.

Reciprocals

Reciprocals in mathematics are numbers which, when multiplied together, result in one. It's like finding the perfect dance partner in math! Each number has a reciprocal. For instance, the reciprocal of 2 is \( \frac{1}{2} \).

• The reciprocal of any given number \( x \) is \( \frac{1}{x} \).
• Zero does not have a reciprocal, as dividing by zero is undefined.
• Numbers and their reciprocals are always related in inverse proportion—when you multiply them, they give a product of 1.

In the sequence \( a_n = \frac{1}{n} \), each term is literally the reciprocal of its position. Learning about reciprocals not only aids with understanding arithmetic sequences but also broadens your math toolkit, as this concept pops up in many areas of math.