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

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New post 15 Jun 2016, 01:46
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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

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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 06 Aug 2016, 06:03
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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:

General Discussion
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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, 03:00
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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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New post 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

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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New post 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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New post 15 Jun 2016, 03:49
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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



\(a_n = (a_1)(a_2)........(a_{n-1})\) = 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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New post 16 Jul 2017, 08:24
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3
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, 22:32
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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


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

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New post 05 Sep 2017, 18:06
1
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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New post 13 Oct 2017, 19:28
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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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New post 15 Nov 2017, 13:49
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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

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, 05:58
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, 06:00
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, 06:39
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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Re: A sequence of numbers a1, a2, a3,…. is defined as follows: a1 = 3, a2  [#permalink]

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New post 20 Mar 2018, 22:33
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


an = t = .......(an - 1)

(an + 1) = .......(an - 1) * an = t^2

(an + 2) = .....(an - 1) * an * (an + 1)

=) t^2 * t ^2 = t^4 = D the answer

thanks
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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 05 May 2018, 03:14
Sash143 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


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



Hello pushpitkc, hope you are having an awesome gmat weekend :)

i have a few questons regarding the above solution, so let me break it down in into following clauses :)

1.) it says "Say n = 3 [given n>2]. Hence we must find value of a5" my question what does \(n\) mean is it the last term ? based on which rule do we concude " If n = 3 [given n>2]. Hence we must find value of \(a_5\) i mean why if n = 3 then we need to find \(a_5\) and not for example \(a_6\) ? :?


2.) if \(a_3 = a_1*a_2 = 3*5 = 15\), then to find \(a_4\) we need to perform following \(a_1*a_2*a_3 = 3*5*15 = 225\)

and following this logic in order to find \(a_5\)we need to do this \(a_1*a_2*a_3*a_4 = 3*5*15*225 = 3375\)

i cant understand why in the solution above to find \(a_5\) we multiply by the same numbers :? \(a_5= 15*15*15*15 = t^4\), to find the next term shoudlnt we mupltiply by all previous terms :?

3.) Also based on this solution \(a_5= 15*15*15*15 = t^4\) why the correct answer is \(t^4\) because there are FOUR numbers 15 :? ? but there could be infinite NUMBERS of 15 :?

And the last question :-) is it geometric sequence question ?

many thanks for taking time to explain :)

enjoy the weekend :)
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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 03 Oct 2018, 04:53
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For problems like this, it’s best that we observe the first few terms of the sequence to see if there’s any noticeable pattern among the results.

Since it’s given that \(a_1=3\) and \(a_2=5\), this means that

\(a_3=(a_1 )(a_2 )=3(5)=15=15^1\)
\(a_4=(a_1 )(a_2 )(a_3 )=(3)(5)(15)=225=15^2\)
\(a_5=(a_1 )(a_2 )(a_3 )(a_4 )=(3)(5)(15)(15^2 )= 15^4 =(15^2 )^2\)
\(a_6=(3)(5)(15)(15^2 )(15^4 )= 15^8=(15^4 )^2\)

As can be seen, as the sequence progresses, the next term would be the square of the term before it.

So, if we start at \(a_n=t\), the next term would be \(t^2\) and term after that would be \((t^2 )^2=t^4\).

\(a_n=t\)
\(a_{(n+1)}=t^2\)
\(a_{(n+2 )}=(t^2 )^2=t^4\)

Hence, the final answer is .
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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 03 Oct 2018, 05:14
EMPOWERMathExpert wrote:
For problems like this, it’s best that we observe the first few terms of the sequence to see if there’s any noticeable pattern among the results.

Since it’s given that \(a_1=3\) and \(a_2=5\), this means that

\(a_3=(a_1 )(a_2 )=3(5)=15=15^1\)
\(a_4=(a_1 )(a_2 )(a_3 )=(3)(5)(15)=225=15^2\)
\(a_5=(a_1 )(a_2 )(a_3 )(a_4 )=(3)(5)(15)(15^2 )= 15^4 =(15^2 )^2\)
\(a_6=(3)(5)(15)(15^2 )(15^4 )= 15^8=(15^4 )^2\)

As can be seen, as the sequence progresses, the next term would be the square of the term before it.

So, it we start at \(a_n=t\), the next term would be \(t^2\) and term after that would be \((t^2 )^2=t^4\).

\(a_n=t\)
\(a_{(n+1)}=t^2\)
\(a_{(n+2 )}=(t^2 )^2=t^4\)

Hence, the final answer is .



Hey EMPOWERMathExpert :) i have been waiting for your explanation for many years:-) what took you so long :lol: welcome to Club of forever students :lol:
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Re: A sequence of numbers a1, a2, a3,…. is defined as follows: a1 = 3, a2 &nbs [#permalink] 03 Oct 2018, 05:14
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