Consider the arithmetic sequence 67, 63, 59,..., 3, -1. How many terms are there in this sequence?单项选择题
A
A. 17
B
B. 19
C
C. 16
D
D. 18

Question

Consider the arithmetic sequence 67, 63, 59,..., 3, -1. How many terms are there in this sequence?

BlackTom AI · Accepted Answer

The common difference is -4, so use the arithmetic-sequence formula a_n = a_1 + (n - 1)d.

1. Relevant knowledge points and concepts
- An arithmetic sequence has a constant difference d between consecutive terms.
- The nth term formula: a_n = a_1 + (n - 1)d.
- To count terms between two values inclusive, you can divide the total change by the step size (absolute value) to get the number of steps, then add 1 for the number of terms.

2. Step-by-step reasoning
- Identify the first term a_1 = 67 and the last term given a_n = -1.
- Compute the common difference: 63 - 67 = -4 (so d = -4).
- Plug into the nth-term formula: -1 = 67 + (n - 1)(-4).

3. Calculations (all intermediate steps)
- Start: -1 = 67 - 4(n - 1)
- Move 67 to left: -1 - 67 = -4(n - 1)
- Simplify left: -68 = -4(n - 1)
- Divide both sides by -4: (-68)/(-4) = n - 1
- Simplify: 17 = n - 1
- Solve for n: n = 17 + 1 = 18

Alternate check using steps: total change from 67 down to -1 is 67 - (-1) = 68. Each step changes value by 4, so number of steps = 68 / 4 = 17 steps. Number of terms = steps + 1 = 18.

4. Why the correct answer is correct
The algebraic solution and the step-counting check both show there are exactly 18 terms required for the sequence to go from 67 to -1 by increments of -4. This matches the arithmetic formula result.

5. Why the other options are incorrect
- A (17): 17 would be the number of steps between first and last terms, not the number of terms; off by one.
- B (19): too many; would correspond to 18 steps or a total change of 72, which is not the case.
- C (16): too few; would correspond to only 15 steps, insufficient to reach -1 from 67.

6. Conclusion
Therefore the sequence has 18 terms; the correct choice is D. 18.

Consider the arithmetic sequence 67, 63, 59,..., 3, -1. How many terms are there in this sequence?单项选择题

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