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

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15 Jun 2016, 00:46
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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 [?]: 139373 [6], given: 12787

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, 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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15 Jun 2016, 02:10
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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

Kudos [?]: 1292 [3], given: 910

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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, 02:27
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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, 02:49
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

$$a_n = (a_1)(a_2)........(a_{n-1})$$ = 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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GMAT 1: 780 Q51 V46
Re: A sequence of numbers a1, a2, a3,…. is defined as follows: a1 = 3, a2 [#permalink]

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06 Aug 2016, 05:03
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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, 07:24
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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, 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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05 Sep 2017, 17:06
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Expert's post
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, 18:28
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

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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15 Nov 2017, 12:49
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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

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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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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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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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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"Be challenged at EVERY MOMENT."

“Strength doesn’t come from what you can do. It comes from overcoming the things you once thought you couldn’t.”

"Each stage of the journey is crucial to attaining new heights of knowledge."

Kudos [?]: 1466 [0], given: 447

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