The following sequence consists of a block of 6 numbers that repeats indefinitely: 1, 4, 2, 5, 7, 8, 1, 4, 2, 5, 7, 8, ... Which of the following is the 65th term of the sequence?
7 is the 65th term of the sequence.
The sequence consists of a repeating block of six numbers: 1, 4, 2, 5, 7, and 8. To find the 65th term, we calculate the position within the repeating block by taking the remainder of 65 divided by 6, which is 5. Therefore, the 65th term corresponds to the fifth number in the sequence, which is 7.
The number 2 is the third number in the repeating sequence (1, 4, 2), and it corresponds to the 3rd position in the block. Since the 65th term is the 5th position, this choice is incorrect.
The number 5 appears as the fourth number in the repeating sequence (1, 4, 2, 5). This corresponds to the 4th position in the block. As the 65th term aligns with the 5th position, this option does not match.
The number 7 is the fifth number in the repeating sequence (1, 4, 2, 5, 7). Since the calculation for the 65th term shows it corresponds to the 5th position of the block, this choice is correct.
The number 8 is the sixth number in the repeating sequence (1, 4, 2, 5, 7, 8), which corresponds to the 6th position in the block. Therefore, this does not align with the 65th term, which is in the 5th position.
The 65th term of the sequence is determined by the repeating pattern of six numbers. By calculating the position of the 65th term within this cycle, it is found to be the 5th number, which is 7. Thus, 7 is the correct answer, reinforcing how modular arithmetic can effectively solve problems involving repeating sequences.
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