To solve this problem, the best choice is an explicit formula. Here's why:

1. Describe the pattern using words:

• The sequence increases by 2 each time, starting at 2. This description is accurate but impractical for finding the 50th term. Calculating the 50th term by repeatedly adding 2 would be time-consuming and prone to error.

2. Use a recursive formula:

• A recursive formula for this sequence would be: $a_1 = 2 \quad \text{and} \quad a_n = a_{n-1} + 2 \text{ for } n \geq 2.$ While this is valid, it requires knowing all preceding terms to compute $a_{50}$. This approach is inefficient, especially for large terms.

3. Use an explicit formula:

• The explicit formula for the sequence is: $a_n = 2n.$ This allows direct computation of any term without needing to find previous ones. For the 50th term: $a_{50} = 2(50) = 100.$ This is the most efficient and practical approach.

Why the explicit formula is the best:

• It is concise, generalizable, and directly computes any term of the sequence, making it ideal for Hilary’s test question. Unlike the other methods, it avoids unnecessary intermediate steps and minimizes the risk of error.

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Final Answer:

The explicit formula, $a_n = 2n$, is the best choice because it directly and efficiently computes the 50th term: $a_{50} = 100$.

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Follow-Up Questions:

1. What are the advantages of explicit formulas in real-world applications?
2. How can you derive an explicit formula from a given sequence?
3. How does the recursive formula differ from the explicit formula in terms of efficiency?
4. Could this sequence be classified as arithmetic? Why or why not?
5. How would the formulas change if the first term were different?

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Tip:

When analyzing sequences, always determine whether an explicit formula can be derived—it’s typically the most efficient solution for finding specific terms.