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

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

by rsarashi » Thu Apr 20, 2017 9:40 am
A sequence of numbers a1, a2, a3,.... is defined as follows: a1=3a1=3, a_2 = 5, and every term in the sequence after a2a2 is the product of all terms in the sequence preceding it, e.g, a3=(a1)(a2)a3=(a1)(a2) and a4=(a1)(a2)(a3)a4=(a1)(a2)(a3). If an=tan=t and n>2n>2, what is the value of an+2an+2 in terms of t?

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

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by DavidG@VeritasPrep » Thu Apr 20, 2017 9:46 am
A sequence of numbers a1,a2,a3,.... is defined as follows: a1 = 3, a2 = 5, and every term in the sequence after a2 is the product of all terms in the sequence preceding it, e.g, a3=(a1)(a2) and a4=(a1)(a2)(a3). If an=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

OA:D
Say n = 3. A3 = 3*5 = 15, so A3 = t = 15.

We also know that A4 = 3*5*15 = 15*15 (the product of the three previous terms.

To summarize: A1 = 3, A2 = 5; A3 =15 and A4 =15*15

If n = 3, n + 2 would be 5. A5= 3 * 5 * 15 *(15*15); combine the 3 and the 5 to get 15*15*15*15 = 15^4

So if A3 = t = 15, and A5 = 15^4, then the answer is D
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by Jay@ManhattanReview » Thu Apr 20, 2017 9:04 pm
rsarashi wrote:A sequence of numbers a1,a2,a3,.... is defined as follows: a1 = 3, a2 = 5, and every term in the sequence after a2 is the product of all terms in the sequence preceding it, e.g, a3=(a1)(a2) and a4=(a1)(a2)(a3). If an=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

OA:D
We have a1 = 3 and a2 = 5.

We have to find out a(n+2) = ??t = ??an

Since n > 2, the least value of n is 3. This would make us compute a(n+2) = a5, which is manageable

Let us first compute t = an = a3

a3 = a1*a2 = 3*5 = t

a5 = a1*a2*a3*a4 = 3*5*(3*5)*(3*5*3*5) = 3^4*5^4

Thus, [spoiler]a5 = 3^4*5^4 = (3*5)^4 = t^4[/spoiler]

The correct answer: D

Hope this helps!

Relevant book: Manhattan Review GMAT Number Properties Guide

-Jay
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