Problem 14 - Algebra and Trigonometry

Question

Find the first five terms of the given recursively defined sequence. \(a_{n}=\frac{1}{1+a_{n-1}} \quad\) and \(\quad a_{1}=1\)

Step-by-Step Solution

Answer

The first five terms are 1, \(\frac{1}{2}\), \(\frac{2}{3}\), \(\frac{3}{5}\), \(\frac{5}{8}\).

Step 1: Determine the First Term

Given that the first term of the sequence is \(a_1 = 1\). This value is provided directly, so the first term is simply \(a_1 = 1\).

Step 2: Calculate the Second Term

Using the recursive formula \(a_n = \frac{1}{1 + a_{n-1}}\) and the known value \(a_1 = 1\), calculate the second term:\[a_2 = \frac{1}{1 + a_1} = \frac{1}{1 + 1} = \frac{1}{2}\]

Step 3: Calculate the Third Term

Using the recursive formula with \(a_2\):\[a_3 = \frac{1}{1 + a_2} = \frac{1}{1 + \frac{1}{2}} = \frac{1}{\frac{3}{2}} = \frac{2}{3}\]

Step 4: Calculate the Fourth Term

Using the recursive formula with \(a_3\):\[a_4 = \frac{1}{1 + a_3} = \frac{1}{1 + \frac{2}{3}} = \frac{1}{\frac{5}{3}} = \frac{3}{5}\]

Step 5: Calculate the Fifth Term

Using the recursive formula with \(a_4\):\[a_5 = \frac{1}{1 + a_4} = \frac{1}{1 + \frac{3}{5}} = \frac{1}{\frac{8}{5}} = \frac{5}{8}\]

Key Concepts

Sequence Terms

In mathematics, a sequence is simply an ordered list of numbers. Each number in this list is called a "term". When dealing with recursive sequences, each term of the sequence is generated based on the previous term or terms. This allows you to construct an infinite sequence from a starting point by repeating the same process over and over.

To better understand sequence terms, consider the recursive sequence in the exercise: starting with the first term, which is given as \(a_1 = 1\), each following term is calculated using the recurrence relation. For this particular sequence, the first five terms are: \(a_1 = 1\), \(a_2 = \frac{1}{2}\), \(a_3 = \frac{2}{3}\), \(a_4 = \frac{3}{5}\), and \(a_5 = \frac{5}{8}\).

When working with sequence terms, make sure:

• To know the initial term or terms (often provided in problems as \(a_1, a_2,\) etc.).
• To consistently apply the recursive formula to find subsequent terms.

Recurrence Relation

The concept of a recurrence relation is essential to understanding recursive sequences. A recurrence relation is an equation that defines each term of a sequence as a function of the preceding terms. It acts like a blueprint for building the sequence.

In the exercise, the recurrence relation is given by:

• \(a_n = \frac{1}{1+a_{n-1}}\)
• This formula tells us how to find the \(n\)-th term if we know the \((n-1)\)-th term.

To utilize a recurrence relation effectively:

• Start from the initial determined term or terms.
• Apply the formula step-by-step to find the desired number of terms.
• Ensure careful substitution, especially with fractions, to avoid mistakes.

Algebra Problem Solving

Algebraic problem-solving skills are crucial when working with recursive sequences. These skills allow you to manipulate equations and expressions to find the solution systematically.

The solution of the given problem showcases the algebra skills needed:

• Understanding how to substitute known terms back into the recurrence relation to find new terms.
• Handling fractions and simplifications, such as \(\frac{1}{1+\frac{1}{2}} = \frac{1}{3/2} = \frac{2}{3}\).
• Rewriting complex fractions into simpler forms by inversing and multiplying them.

Algebra helps in simplifying and accurately calculating each term, ensuring the progression follows the correct pattern. Mastering these algebraic techniques makes solving recursive sequences more straightforward and helps develop deeper math problem-solving skills.