Question

Define the following sequence recursively, giving two instructions in mathematical language.

2, 6, 10, 14,...)
a. a₁ = 1, an
an-1+4, for n ≥2
b. a₁ = 2,an = an-1+4, for n ≥ 2
X c. a₁ = 2,an = an-1+4, for n ≤2
d. a₁ = 2,an = an-1-4, for n≥2

Answer 1

The sequence 2, 6, 10, 14, ... is defined recursively as a1 = 2 and an = an-1 + 4 for n ≥ 2, creating an arithmetic sequence with a common difference of 4. So the correct answer is Option A.

Step-by-step explanation:

The student is asking about how to recursively define a given sequence. The sequence provided is 2, 6, 10, 14, ..., which represents an arithmetic sequence where each term increases by 4 from the previous term. The correct recursive definition for this sequence would be:

• a1 = 2, this is the initial term of the sequence.
• an = an-1 + 4, for n ≥ 2, which means that each term after the first is the previous term plus 4.

By using this recursive formula, we can generate the terms of the given sequence by starting with the first term and then repeatedly applying the second instruction to find each subsequent term.