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A sequence of numbers a1, a2, a3,…. is defined as follows: a1 = 3, a2

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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: a1 = 3, a2 [#permalink]

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New post 15 Jun 2016, 02:00
IMO D

An=t so that means product from A1 to An-1 is t. therefore An+1 = A1x.....An-1 x An=txt= t^2
proceeding in same way An+2 will be t^4

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Re: A sequence of numbers a1, a2, a3,…. is defined as follows: a1 = 3, a2 [#permalink]

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An = (A1)(A2)......(An-1) = t
An+1 = t * An = t^2
An+2 = t * An * An+1 = t * t * t^2 = t^4

Answer: D

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Re: A sequence of numbers a1, a2, a3,…. is defined as follows: a1 = 3, a2 [#permalink]

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Bunuel 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



Given \(a_n = t\)
This means \(a_1 * a_2 * a_3* ..... a_{n-1} = t\)

Therefore \(a_{n+1} = (a_1 * a_2 * a_3* ..... a_{n-1}) * (a_n) = t*t = t^2\)

and \(a_{n+2} = (a_1 * a_2 * a_3* ..... a_{n-1}) * (a_n) * (a_{n+1}) = t*t*t^2 = t^4\)

Answer = \(D\)

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Re: A sequence of numbers a1, a2, a3,…. is defined as follows: a1 = 3, a2 [#permalink]

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Bunuel 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



\(a_n = (a_1)(a_2)........(a_{n-1})\) = t

\(a_{n+1}\) = \((a_1)(a_2)........(a_n)\) = \((a_1)(a_2)........(a_{n-1})\)*\(a_n\) = t *t = \(t^2\)

\(a_{n+2}\) = \((a_1)(a_2)........(a_{n+1})\) = \((a_1)(a_2)........(a_{n-1})\) * \(a_n\) *\(a_{n+1}\)

= t * t * \(t^2\) = \(t^4\)

D is the answer.

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Re: A sequence of numbers a1, a2, a3,…. is defined as follows: a1 = 3, a2 [#permalink]

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Here are a couple of ways to think about this problem:

OG 17 q201 Method 1:



OG 17 q201 Method 2:

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Re: A sequence of numbers a1, a2, a3,…. is defined as follows: a1 = 3, a2 [#permalink]

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Bunuel 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


Say n = 3 [given n>2]. Hence we must find value of a5

a1 = 3,
a2 = 5,
a3 = 5*3 = 15 = t [an = t given]
a4 = 15*15
a5= 15*15*15*15 = t^4
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Re: A sequence of numbers a1, a2, a3,…. is defined as follows: a1 = 3, a2 [#permalink]

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New post 29 Aug 2017, 21:32
Bunuel 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


\(a_3 = a_n = a_1*a_2 = t\)
\(a_4 = a_{n + 1} = a_1*a_2*a_3 = t*a_3 = t*t = t^2\)
\(a_5 = a_{n + 2} = a_1*a_2*a_3*a_4 = t*t*(t^2) = t^4\)
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Re: A sequence of numbers a1, a2, a3,…. is defined as follows: a1 = 3, a2 [#permalink]

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Bunuel 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


We are given a sequence in which every term in the sequence after a(2) is the product of all terms in the sequence preceding it. So:

a(n+1) = a(n) x a(n-1) x ... x a(2) x a(1)

By the same reasoning, we have:

a(n) = a(n-1) x a(n-2) x ... x a(2) x a(1)

We can substitute a(n-1) x... x a(2) x a(1) in the a(n+1) equation for a(n), so we have a(n+1) = a(n) x a(n).

However, recall that a(n) = t, so a(n+1) = t x t = t^2. By the same reasoning, we have:

a(n+2) = a(n+1) x a(n) x a(n-1) x ... x a(2) x a(1)

However, a(n) x a(n-1) x .... x a(2) x a(1) = a(n+1) and a(n+1) = t^2, so:

a(n+2) = a(n+1) x a(n+1) = t^2 x t^2 = t^4

Answer: D
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A sequence of numbers a1, a2, a3,…. is defined as follows: a1 = 3, a2 [#permalink]

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Bunuel 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


Observe/Create a sequence:
A3 = A1*A2
A4 = A3*A1*A2 = A3*A3
A5 = A4*A4
.
.
.
An = t
An+1 = An*An = t*t = t^2
An+2 = An+1*An+1 = t^2*t^2 = t^4

D.

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Re: A sequence of numbers a1, a2, a3,…. is defined as follows: a1 = 3, a2 [#permalink]

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Bunuel 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: a1 = 3, a2 [#permalink]

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New post 02 Jan 2018, 04:58
Bunuel 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


Bunuel - can I call the above mentioned problem as a recursive sequence ? :? I know this formula An= n1+(n-1)*d but could not apply it here.

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Re: A sequence of numbers a1, a2, a3,…. is defined as follows: a1 = 3, a2 [#permalink]

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New post 02 Jan 2018, 05:00
hazelnut wrote:
Bunuel 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


\(a_3 = a_n = a_1*a_2 = t\)
\(a_4 = a_{n + 1} = a_1*a_2*a_3 = t*a_3 = t*t = t^2\)
\(a_5 = a_{n + 2} = a_1*a_2*a_3*a_4 = t*t*(t^2) = t^4\)



Hello hazelnut - how did you figure out that A_n+2 is the fifth term and not sixth :? ?

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A sequence of numbers a1, a2, a3,…. is defined as follows: a1 = 3, a2 [#permalink]

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New post 02 Jan 2018, 05:39
dave13 wrote:
hazelnut wrote:
Bunuel 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


\(a_3 = a_n = a_1*a_2 = t\)
\(a_4 = a_{n + 1} = a_1*a_2*a_3 = t*a_3 = t*t = t^2\)
\(a_5 = a_{n + 2} = a_1*a_2*a_3*a_4 = t*t*(t^2) = t^4\)


Hello hazelnut - how did you figure out that A_n+2 is the fifth term and not sixth :? ?


let n = 3. \(a_3 = a_n = a_1*a_2 = t\)
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Rules for posting in verbal forum | Please DO NOT post short answer in your post!

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A sequence of numbers a1, a2, a3,…. is defined as follows: a1 = 3, a2   [#permalink] 02 Jan 2018, 05:39
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