Problem 41 - Algebra and Trigonometry

Question

A sequence is defined recursively. List the first five terms. \(a_{1}=3 ; \quad a_{n}=\frac{a_{n-1}}{n}\)

Step-by-Step Solution

Answer

The first five terms are 3, 1.5, 0.5, 0.125, and 0.025.

Step 1: Identify the First Term

The first term of the sequence is given as \(a_{1} = 3\).

Step 2: Calculate the Second Term

Use the recursive formula \(a_{n} = \frac{a_{n-1}}{n}\) to find the second term. For \(n = 2\), we have \(a_{2} = \frac{a_{1}}{2} = \frac{3}{2}\).

Step 3: Calculate the Third Term

Now, calculate the third term using the recursive formula. For \(n = 3\), \(a_{3} = \frac{a_{2}}{3} = \frac{\frac{3}{2}}{3} = \frac{3}{2 \times 3} = \frac{1}{2}\).

Step 4: Calculate the Fourth Term

Next, find the fourth term. For \(n = 4\), \(a_{4} = \frac{a_{3}}{4} = \frac{\frac{1}{2}}{4} = \frac{1}{2 \times 4} = \frac{1}{8}\).

Step 5: Calculate the Fifth Term

Finally, compute the fifth term. For \(n = 5\), \(a_{5} = \frac{a_{4}}{5} = \frac{\frac{1}{8}}{5} = \frac{1}{8 \times 5} = \frac{1}{40}\).

Key Concepts

sequence definition

To understand the problem, we need to grasp what a sequence is. A sequence is simply a list of numbers arranged in a specific order. Each number in the sequence is called a term. In this case, we have a **recursive sequence**. A recursive sequence is one where each term is defined based on its position and the terms before it. The starting point is known as the **first term**, and subsequent terms are determined by a rule or formula. In our exercise, the first term is given as \( a_1 = 3 \), and the rule to find the next terms is provided by the recursive formula.

recursive formula

A **recursive formula** is a rule that relates each term of a sequence to its predecessors. It's like a recipe that tells you how to cook the next dish using the previous one. For the sequence in our exercise, the recursive formula is \( a_n = \frac{a_{n-1}}{n} \). This means that to find the \( n \)-th term, you need to divide the previous term by \( n \).

Here's how this works:

• For \( n = 2 \), the term \( a_2 \) is obtained by dividing the first term \( a_1 \) by 2.
• Similarly, for \( n = 3 \), the term \( a_3 \) is found by dividing \( a_2 \) by 3.
• This process continues for each subsequent term.

sequence terms calculation

Now that we understand the sequence definition and the recursive formula, we can calculate the first five terms:

• First Term: This is given directly as \( a_1 = 3 \).
• Second Term: Using the recursive formula \( a_2 = \frac{a_1}{2} = \frac{3}{2} \).
• Third Term: Again, apply the recursive formula: \( a_3 = \frac{a_2}{3} = \frac{\frac{3}{2}}{3} = \frac{3}{2 \times 3} = \frac{1}{2} \).
• Fourth Term: \( a_4 = \frac{a_3}{4} = \frac{\frac{1}{2}}{4} = \frac{1}{2 \times 4} = \frac{1}{8} \).
• Fifth Term: Finally, \( a_5 = \frac{a_4}{5} = \frac{\frac{1}{8}}{5} = \frac{1}{8 \times 5} = \frac{1}{40} \).

By following the steps above, you can calculate any term in a recursive sequence where the first term and the formula are known.