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

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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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Joined: 12 Sep 2015
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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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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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23
3
Here are a couple of ways to think about this problem:

OG 17 q201 Method 1:

OG 17 q201 Method 2:

General Discussion
Manager  G
Joined: 04 Apr 2015
Posts: 104
Re: A sequence of numbers a1, a2, a3,…. is defined as follows: a1 = 3, a2  [#permalink]

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1
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
Marshall & McDonough Moderator 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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5
4
An = (A1)(A2)......(An-1) = t
An+1 = t * An = t^2
An+2 = t * An * An+1 = t * t * t^2 = t^4

Intern  Joined: 29 Mar 2016
Posts: 5
Re: A sequence of numbers a1, a2, a3,…. is defined as follows: a1 = 3, a2  [#permalink]

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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 = $$D$$
Manager  Joined: 18 Jan 2010
Posts: 246
Re: A sequence of numbers a1, a2, a3,…. is defined as follows: a1 = 3, a2  [#permalink]

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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

$$a_n = (a_1)(a_2)........(a_{n-1})$$ = 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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Posts: 79
Re: A sequence of numbers a1, a2, a3,…. is defined as follows: a1 = 3, a2  [#permalink]

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4
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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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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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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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

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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Joined: 09 Mar 2016
Posts: 1230
Re: A sequence of numbers a1, a2, a3,…. is defined as follows: a1 = 3, a2  [#permalink]

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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

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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Joined: 09 Mar 2016
Posts: 1230
Re: A sequence of numbers a1, a2, a3,…. is defined as follows: a1 = 3, a2  [#permalink]

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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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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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Joined: 24 Mar 2015
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Re: A sequence of numbers a1, a2, a3,…. is defined as follows: a1 = 3, a2  [#permalink]

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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

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
VP  D
Joined: 09 Mar 2016
Posts: 1230
A sequence of numbers a1, a2, a3,…. is defined as follows: a1 = 3, a2  [#permalink]

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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 Intern  B
Joined: 15 Sep 2018
Posts: 31
A sequence of numbers a1, a2, a3,…. is defined as follows: a1 = 3, a2  [#permalink]

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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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Posts: 1230
Re: A sequence of numbers a1, a2, a3,…. is defined as follows: a1 = 3, a2  [#permalink]

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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 welcome to Club of forever students Intern  B
Joined: 03 Sep 2018
Posts: 19
WE: Analyst (Consulting)
Re: A sequence of numbers a1, a2, a3,…. is defined as follows: a1 = 3, a2  [#permalink]

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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

$$a_{n+2}$$ = $$a_{n+1}$$ * $$a_{n}$$ * $$a_{n-1}$$ * $$a_{n-2}$$ _ _ _ _ $$a_{1}$$

now also we can write $$a_{n+1}$$ = $$a_{n}$$ * $$a_{n-1}$$ * $$a_{n-2}$$ _ _ _ _ $$a_{1}$$, so substituting this in the above equation

$$a_{n+2}$$ = ($$a_{n}$$ * $$a_{n-1}$$ * $$a_{n-2}$$ _ _ _ _ $$a_{1}$$) ^ 2

again, $$a_{n}$$ = $$a_{n-1}$$ * $$a_{n-2}$$ _ _ _ _ $$a_{1}$$ , substituting above

$$a_{n+2}$$ = (($$a_{n}$$ * $$a_{n}$$)) ^ 2

we are given that $$a_n =t$$, so substituting above

$$a_{n+2}$$ = (t^2)^2 = t^4

So, option D is correct
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