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Vera and Fenyang were asked to find an explicit formula for the sequence , where the first term should be .Vera said the formula is .Fenyang said the formula is .Which one of them is right?Choose answer:(A) Only Vera(B) Only Fenyang(C) Both Vera and Fenyang(D) Neither Vera nor Fenyang
Full solution
Q. Vera and Fenyang were asked to find an explicit formula for the sequence , where the first term should be .Vera said the formula is .Fenyang said the formula is .Which one of them is right?Choose answer:(A) Only Vera(B) Only Fenyang(C) Both Vera and Fenyang(D) Neither Vera nor Fenyang
- Identify pattern: Identify the pattern in the sequence to determine if it is arithmetic or geometric. The sequence decreases by a constant amount each time, which indicates it is an arithmetic sequence.
- Calculate common difference: Calculate the common difference of the sequence by subtracting any term from the previous term. For example, , and . The common difference is .
- Use explicit formula: Use the explicit formula for an arithmetic sequence, which is , where is the first term and is the common difference. The first term is , and the common difference is .
- Substitute values for Vera: Substitute the values of and into the formula to find Vera's proposed formula. Vera's formula is . Let's expand this to check if it matches the sequence: .
- Check Vera's formula: Now, let's check Fenyang's formula, which is . This formula is already in the form of an arithmetic sequence formula, and it matches the expanded form of Vera's formula.
- Both correct: Since both Vera's and Fenyang's formulas simplify to the same expression, they are both correct. Therefore, the answer is that both Vera and Fenyang are right.
