[email protected] wrote:In the sequence 1, 2, 2, ..., An, An = A(n-1) "¢ A(n-2). The value of Aâ‚�₃ is how many times the value of Aâ‚�â‚�?
(A) 2
(B) 2³
(C) 2³²
(D) 2��
(E) 2��
The answer choices should read as I've posted them here.
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.
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