Identifying the patern

Uva@90 wrote:In the sequence 1, 2, 2, …, an, …, an = an-1 • an-2. The value of a13 is how many
times the value of a11?
(A) 2
(B) 23
(C) 232
(D) 264
(E) 289

OA E

Normally for this kind of the questions we need to find the pattern and solve it.

But I don't get any pattern after seeing the question.

Regards,
Uva.

Those answer choices look weird! Could you confirm that this question does not have any typo?

In the sequence 1, 2, 2, …, An, An = A(n-1) • A(n-2). The value of A�₃ is how many times the value of A��?

(A) 2
(B) 2³
(C) 2³²
(D) 2��
(E) 2��

As Rich has noted, the answer choices should read as I've posted them above.

Write it out and LOOK FOR A PATTERN.
Phrase the values in the sequence as the answer choices are phrased: in terms of POWERS OF 2.

A� = 1 = 2�.
A₂ = 2¹.
A₃ = A₂ * A� = 2¹ * 2� = 2¹.
A₄ = A₃ * A₂ = 2*2 = 2².
A₅ = A₄ * A₃ = 2² * 2 = 2³.
A₆ = A₅ * A₄ = 2³ * 2² = 2�.
A₇ = A₆ * A₅ = 2� * 2� = 2�.

Notice the pattern exhibited by the resulting exponents:
Each exponent is equal to the SUM OF THE TWO PRECEDING EXPONENTS.
To illustrate:
The exponent for A₆ (5) is equal to the sum of the exponents for A₅ and A₄ (3 and 2).
The exponent for A₇ (8) is equal to the sum of the exponents for A₆ and A₅ (5 and 3).

Thus, the sequence will proceed as follows:
Exponent for A₈ = sum of the exponents for A₇ and A₆ = 8+5 = 13.
Exponent for A₉ = sum of the exponents for A₈ and A₇ = 8+5 = 13 + 8 = 21.
Exponent for A�₀ = sum of the exponents for A₉ and A₈ = 21 + 13 = 34.
Exponent for A�� = sum of the exponents for A�₀ and A₉ = 34 + 21 = 55.
Exponent for A�₂ = sum of the exponents for A�� and A�₀ = 855+34 = 89.
Exponent for A�₃ = sum of the exponents for A�₂ and A�� = 89 + 55 = 144.

Thus:
A�₃/A��= 2¹��/2�� = 2��.

The correct answer is E.