A sequence {an} is defined as follows:

a0=3,

a1=1, and

an= (an−1⋅an−2 − 2). for n≥2,

What is a3?

Question

A sequence {an} is defined as follows:

a0=3,

a1=1, and

an= (an−1⋅an−2 − 2).  for n≥2,

What is a3?

BlackTom AI · Accepted Answer

Use the recurrence an = (an−1 · an−2 − 2) with the given initial terms to compute successive terms.

Relevant concepts: a linear recurrence here uses the two immediately preceding terms: to find a2 you need a1 and a0; to find a3 use a2 and a1. The formula multiplies the two previous terms and then subtracts 2.

Step-by-step calculation:
1. Known values: a0 = 3, a1 = 1.
2. Compute a2 using n = 2:
   a2 = a1 · a0 − 2
      = 1 · 3 − 2
      = 3 − 2
      = 1.
   (All intermediate arithmetic shown: multiplication 1·3 = 3, then subtraction 3−2 = 1.)
3. Compute a3 using n = 3:
   a3 = a2 · a1 − 2
      = 1 · 1 − 2
      = 1 − 2
      = −1.
   (Intermediate steps: multiplication 1·1 = 1, then 1−2 = −1.)

Why this is correct: the recurrence explicitly requires the product of the two previous terms minus 2. Using the computed a2 = 1 and the given a1 = 1 satisfies the recurrence exactly and yields −1.

Why the other choices are incorrect:
- 1: This would be a2, not a3; a3 computed correctly is −1.
- 3: This is a0, not a3; it would arise from misreading indices.
- 9: No intermediate multiplication yields 9 with these initial values under this recurrence; this likely stems from an arithmetic error (e.g., squaring 3 incorrectly applied).

Conclusion: Following the recurrence and showing each intermediate step gives a3 = −1, which matches the provided correct answer.

A sequence {an} is defined as follows: a0=3, a1=1, and an= (an−1⋅an−2 − 2). for n≥2, What is a3?单项选择题

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