A sequence of numbers $a_1, a_2, a_3,\ldots$ is defined as follows: $a_1=3, a_2=5,$ and every term in the sequence

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VJesus12: Legendary Member; Posts: 2276; Joined: Sat Oct 14, 2017 6:10 amFollowed by: 3 members

by VJesus12 » Fri Jan 28, 2022 7:24 am

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A sequence of numbers $a_1, a_2, a_3,\ldots$ 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 $a_4=(a1)(a2)(a3).$ 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) $t^8$

Answer: D

Source: Official Guide

Source: Beat The GMAT — Problem Solving | Fri Jan 28, 2022 7:24 am

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Re: A sequence of numbers $a_1, a_2, a_3,\ldots$ is defined as follows: $a_1=3, a_2=5,$ and every term in the sequen

by Brent@GMATPrepNow » Fri Jan 28, 2022 7:37 am

VJesus12 wrote: ↑
Fri Jan 28, 2022 7:24 am
A sequence of numbers $a_1, a_2, a_3,\ldots$ 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 $a_4=(a1)(a2)(a3).$ 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) $t^8$

Answer: D

Source: Official Guide

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

Brent Hanneson - Creator of GMATPrepNow.com

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