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Emeka and Maricel were asked to find an explicit formula for the sequence , where the first term should be .Emeka said the formula is .Maricel said the formula is .Which one of them is right?Choose answer:(A) Only Emeka(B) Only Maricel(C) Both Emeka and Maricel(D) Neither Emeka nor Maricel
Full solution
Q. Emeka and Maricel were asked to find an explicit formula for the sequence , where the first term should be .Emeka said the formula is .Maricel said the formula is .Which one of them is right?Choose answer:(A) Only Emeka(B) Only Maricel(C) Both Emeka and Maricel(D) Neither Emeka nor Maricel
- Identify sequence type: Identify the type of sequence. The sequence has a common difference between consecutive terms, which is . This indicates that it is an arithmetic sequence.
- Use explicit formula: Use the explicit formula for an arithmetic sequence, , where is the first term and is the common difference. For this sequence, is and the common difference, , is .
- Substitute values: Substitute the values of and into the formula to write an expression to describe the sequence. The expression for the sequence is .
- Simplify expression: Simplify the expression. .
- Compare to Emeka and Maricel: Compare the simplified expression to the formulas provided by Emeka and Maricel. Emeka's formula is , which is the unsimplified version of the correct formula. Maricel's formula is , which is the simplified version of the correct formula.
- Determine correct formula: Determine which formula is correct. Both formulas describe the same sequence, even though they look different. Emeka's formula is the unsimplified version, and Maricel's formula is the simplified version. Therefore, both are correct.
