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A recursive sequence is a way of defining a sequence where each term depends on the previous one. It's like building a tower by laying new bricks on top of the ones below. In mathematics, instead of bricks, we use numbers or expressions. In such sequences, a formula bares significant meaning as it tells you how to move from one number to the next. For example, the given sequence uses the formula:

• \(a_{n+1} = \frac{1}{a_n}\)

This means each term after the first is the reciprocal (or inverse) of the previous term. This recursive relationship gives a special pattern, often making it easy to compute subsequent terms if you know the earlier ones. Remember, recursive sequences rely fundamentally on knowing the initial or a few preceding terms to determine the future ones.

The initial term is the first step you take when calculating a recursive sequence. Without it, you would be lost, much like without the starting point on a map. In our example, the initial term is denoted by \(a_1 = 117\).This initial value is essential as it gives you the first building block of the sequence:

• It establishes where you begin the journey of calculation.
• Serves as the input into the recursive formula to get subsequent terms.
• Determines the specific pattern of the sequence.

It's crucial to identify and understand this starting point as every calculation thereafter depends on it. Think of it as the foundation of a house; without it, the whole structure won't stand firm.

Step-by-step calculation in recursive sequences is like following a recipe. You need to know each step to move forward accurately. Here's how to calculate terms for the sequence we have.**First Term**:We start with the initial term, which is given by \(a_1 = 117\). **Second Term**:Using the formula \(a_{n+1} = \frac{1}{a_n}\), substitute \(a_1\) to find \(a_2\):

• \(a_2 = \frac{1}{117}\)

**Third Term**:Applying the formula again with the result we got for \(a_2\):

• \(a_3 = \frac{1}{a_2} = 117\)

**Fourth Term**:Finally, use the recursive relationship to determine \(a_4\):

• \(a_4 = \frac{1}{a_3} = \frac{1}{117}\)

Following these steps carefully ensures you correctly compute the sequence. By breaking down each calculation, you can manage more complex sequences with ease. Consider every step a small, carefully planned move to the right answer.