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Understanding mathematical sequences is key to solving the given exercise. A sequence is a list of numbers arranged in a particular order. Each number in the sequence is called a term. The sequence given in the exercise is 16, 4, 2, \(\sqrt{2}\).

Sequences can be finite or infinite, depending on whether they have a defined end or continue indefinitely. The given sequence appears infinite as it continues with `...`. Let's break it down further:

• The first term, \((a_1)\), is 16.
• The second term, \((a_2)\), is 4.
• The third term, \((a_3)\), is 2.
• The fourth term, \((a_4)\), is \(\sqrt{2}\)).

To understand how sequences work, you need a clear grasp of the order and pattern of the numbers. As you can see, each number directly relates to the one preceding it. This relationship between terms is what you capture using a recursion formula.

In mathematical terms, a recurrence relation expresses each term of the sequence as a function of one or more of its preceding terms. It’s a way to recursively generate the terms of a sequence. The given exercise involves identifying such a relation for the sequence {16, 4, 2, \(\sqrt{2}\).

The steps to derive the recurrence relation are:

• Identify the pattern in the sequence.
• Express this pattern with a mathematical formula.
• Verify the formula by checking if it accurately produces the given terms.

In our sequence, each term is derived by dividing the previous term by a certain number:

• For \((a_2)\): \((4 = 16/4)\)
• For \((a_3)\): \((2 = 4/2)\)
• For \((a_4)\): \(( \sqrt{2} = 2 / \sqrt{2} )\)

This consistent pattern shows that each term (\