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A sequence of numbers a1, a2, a3,…. is defined as follows: a

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A sequence of numbers a1, a2, a3,…. is defined as follows: a [#permalink] New post 30 Jun 2020, 09:15
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A sequence of numbers \(a_1\), \(a_2\), \(a_3\),…. is defined as follows: \(a_1 = 3\), \(a_2 = 5\), and every term in the sequence after \(a_2\) is the product of all terms in the sequence preceding it, e.g, \(a_3 = (a_1)(a_2)\) and \(a4 = (a_1)(a_2)(a_3)\). If \(a_n =t\) and \(n > 2\), what is the value of \(a_{n+2}\) in terms of t?

(A) 4t
(B) t^2
(C) t^3
(D) t^4
(E) t^8
[Reveal] Spoiler: OA

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Re: A sequence of numbers a1, a2, a3,…. is defined as follows: a [#permalink] New post 30 Jun 2020, 09:16
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Re: A sequence of numbers a1, a2, a3,…. is defined as follows: a [#permalink] New post 30 Jun 2020, 09:40
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Carcass wrote:
A sequence of numbers \(a_1\), \(a_2\), \(a_3\),…. is defined as follows: \(a_1 = 3\), \(a_2 = 5\), and every term in the sequence after \(a_2\) is the product of all terms in the sequence preceding it, e.g, \(a_3 = (a_1)(a_2)\) and \(a4 = (a_1)(a_2)(a_3)\). If \(a_n =t\) and \(n > 2\), what is the value of \(a_{n+2}\) in terms of t?

(A) 4t
(B) t^2
(C) t^3
(D) t^4
(E) t^8


Let's list a few terms....
term1 = 3
term2 = 5
term3 = (term2)(term1) = (5)(3) = 15 (term2)(term1)
term4 = (term3)(term2)(term1) = (15)(5)(3) = 15²
term5 = (term4)(term3)(term2)(term1) = (15²)(15)(5)(3) = 15⁴
term6 = (term5)(term4)(term3)(term2)(term1) = (15⁴)(15²)(15)(5)(3) = 15⁸

At this point, we can see the pattern.

Continuing, we get....
term7 = 15^16
term8 = 15^32

Each term in the sequence is equal to the SQUARE of term before it

If term_n =t and n > 2, what is the value of term_n+2 in terms of t?
So, term_n = t
term_n+1 = t²
term_n+2 = t⁴

Answer: D

Cheers,
Brent
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Re: A sequence of numbers a1, a2, a3,…. is defined as follows: a   [#permalink] 30 Jun 2020, 09:40
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