Problem 47 - Algebra and Trigonometry

Question

A sequence is defined recursively. List the first five terms. \(a_{1}=\sqrt{2} ; \quad a_{n}=\sqrt{2+a_{n-1}}\)

Step-by-Step Solution

Answer

The first five terms are \( \sqrt{2} \), \( \sqrt{2 + \sqrt{2}} \), \( \sqrt{2 + \sqrt{2 + \sqrt{2}}} \), \( \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2}}}} \), and \( \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2}}}}} \).

Step 1: Identify the First Term

The first term of the sequence is given as \(a_1 = \sqrt{2}\).

Step 2: Calculate the Second Term

Use the recursive formula \(a_n = \sqrt{2 + a_{n-1}}\) to find the second term: \[a_2 = \sqrt{2 + a_1} = \sqrt{2 + \sqrt{2}}.\]

Step 3: Calculate the Third Term

Continue using the recursive formula to find the third term: \[a_3 = \sqrt{2 + a_2} = \sqrt{2 + \sqrt{2 + \sqrt{2}}}.\]

Step 4: Calculate the Fourth Term

Now, find the fourth term using the same formula: \[a_4 = \sqrt{2 + a_3} = \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2}}}}.\]

Step 5: Calculate the Fifth Term

Finally, calculate the fifth term: \[a_5 = \sqrt{2 + a_4} = \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2}}}}}.\]

Key Concepts

recursive formula

A recursive formula is a way of defining the terms in a sequence by relating each term to previous ones. In this exercise, the recursive formula is given by:

a_{n} = \sqrt{2 + a_{n-1}}.

This means that to find term \(a_{n}\), you take the square root of 2 plus the previous term \(a_{n-1}\).

Using a recursive formula allows you to generate terms step-by-step, which can make the sequence easier to analyze, as each term is built from the preceding one.

This approach can be particularly helpful when dealing with complex sequences, as it breaks down the process into manageable chunks.

sequence terms

The sequence terms are the individual elements of the sequence. In this problem, the first term \(a_1\) is given directly, while the following terms are calculated using the recursive formula. Here are the first five terms for our sequence:

• Term 1: \(a_1 = \sqrt{2}\)
• Term 2: \(a_2 = \sqrt{2 + \sqrt{2}}\)
• Term 3: \(a_3 = \sqrt{2 + \sqrt{2 + \sqrt{2}}}\)
• Term 4: \(a_4 = \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2}}}}\)
• Term 5: \(a_5 = \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2}}}}}}\)

Notice how each term builds upon the previous one, illustrating the concept of recursion, where each term's definition depends on its predecessor.

square root function

The square root function is a mathematical operation that finds a value that, when multiplied by itself, gives the original number. In this exercise, the square root function is used repeatedly. For example, the second term is \(\sqrt{2 + \sqrt{2}}\).

To calculate a term using the square root function:

• Add the constant (2 in this case) to the previous term.
• Then apply the square root to the result of this addition.

The square root function helps in recursively defining the sequence. Because this function is involved in every term, it significantly affects how the sequence grows and progresses.

mathematical sequences

Mathematical sequences are ordered lists of numbers where each number is identified by its position in the list. Sequences can be defined in various ways, including explicitly (giving a formula for the nth term) or recursively, as seen in this exercise.

• Explicit Definition: A direct formula to find any term. For example, \(a_n = n^2\) for a sequence of square numbers.
• Recursive Definition: A formula that defines each term based on previous terms, like our formula \(a_n = \sqrt{2 + a_{n-1}}\).

Understanding sequences is crucial in many areas of mathematics because they often describe patterns and progressions found in real-world contexts as well as theoretical problems.