Question

A sequence {an} is defined recursively, with a1 = 1, a2 = 2, and, for n > 2, an = an-1 * an-2. Find the first seven terms of the sequence.

A) 1, 2, 2, 4, 8, 32, 256
B) 1, 2, 2, 1, 1/2, 1/2, 1
C) 1, 2, 1/2, 1, 2, 1/2, 1
D) 1, 2, 1/2, 4, 1/8, 32, 1/256

Answer 1

By applying the recursive definition an = an-1 * an-2 starting from the initial conditions a1 = 1 and a2 = 2, the first seven terms of the sequence are calculated as 1, 2, 2, 4, 8, 32, 256.

Step-by-step explanation:

The student is asking for the first seven terms of a recursively defined sequence with the initial conditions a1 = 1, a2 = 2, and the recursion relation an = an-1 * an-2 for n > 2. To find the terms of the sequence, we simply apply the recursive formula successively, using the initial conditions given.

• So a3 = a2 * a1 = 2 * 1 = 2
• a4 = a3 * a2 = 2 * 2 = 4
• a5 = a4 * a3 = 4 * 2 = 8
• a6 = a5 * a4 = 8 * 4 = 32
• a7 = a6 * a5 = 32 * 8 = 256

Thus, the first seven terms of the sequence are 1, 2, 2, 4, 8, 32, 256, which matches option A in the question.