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

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

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15 Jun 2016, 01:46
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60% (01:45) correct 40% (01:50) wrong based on 775 sessions

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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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Kudos [?]: 129080 [5], given: 12194

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

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15 Jun 2016, 03:00
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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15 Jun 2016, 03:10
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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

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

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15 Jun 2016, 03:27
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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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15 Jun 2016, 03:49
1
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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 * $$t^2$$ = $$t^4$$

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

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06 Aug 2016, 06:03
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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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16 Jul 2017, 08:24
2
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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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29 Aug 2017, 22:32
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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05 Sep 2017, 18:06
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

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

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13 Oct 2017, 19:28
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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A sequence of numbers a1, a2, a3,…. is defined as follows: a1 = 3, a2   [#permalink] 13 Oct 2017, 19:28
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