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# The infinite sequence a1, a2,...,an is defined such that an

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Manager
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The infinite sequence a1, a2,...,an is defined such that an [#permalink]  15 Nov 2010, 17:19
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The infinite sequence a1, a2,...,an is defined such that an = (n+2) / n for all n ≥ 1. What is the product of the first 10 terms of the sequence?

(A) 45
(B) 66
(C) 90
(D) 121
(E) 132
[Reveal] Spoiler: OA
Manager
Joined: 03 Aug 2010
Posts: 108
GMAT Date: 08-08-2011
Followers: 1

Kudos [?]: 12 [0], given: 63

Re: Sum of consecutive terms [#permalink]  15 Nov 2010, 17:39
krushna wrote:
(n+2)/n..
So the product of first 10 numbers

3 * 4 * 5 *....10*11*12/1*2*3*4*........10
= 11*12/1*2 = 66

I didn't follow the logic or the notation. I'm also not sure what's in parenthesis and what's not.
Veritas Prep GMAT Instructor
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Kudos [?]: 3623 [0], given: 144

Re: Sum of consecutive terms [#permalink]  15 Nov 2010, 17:56
Expert's post
Yalephd wrote:
The infinite sequence a1, a2,...,an is defined such that an = (n+2) / n for all n ≥ 1. What is the product of the first 10 terms of the sequence?

(A) 45
(B) 66
(C) 90
(D) 121
(E) 132

Powering through this would take the full lenght of the GMAT (maybe not). Is there a quick way to solve this problem?

Look at the pattern:
an = \frac{(n + 2)}{n}
So a1 = 3/1
a2 = 4/2
a3 = 5/3
a4 = 6/4

Now when we multiply the first ten terms (n = 1 to 10), \frac{3}{1} * \frac{4}{2} * \frac{5}{3} * \frac{6}{4} * ...* \frac{11}{9} * \frac{12}{10}
denominators and numerators will get canceled, 3 with 3, 4 with 4 etc. Only the first two denominators and last two numerators will be left. The product will be 11*12/2 = 66
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Karishma
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Save $100 on Veritas Prep GMAT Courses And Admissions Consulting Enroll now. Pay later. Take advantage of Veritas Prep's flexible payment plan options. Veritas Prep Reviews Intern Joined: 13 Oct 2010 Posts: 20 Followers: 0 Kudos [?]: 5 [0], given: 0 Re: Sum of consecutive terms [#permalink] 09 Dec 2010, 08:12 VeritasPrepKarishma wrote: Yalephd wrote: The infinite sequence a1, a2,...,an is defined such that an = (n+2) / n for all n ≥ 1. What is the product of the first 10 terms of the sequence? (A) 45 (B) 66 (C) 90 (D) 121 (E) 132 Powering through this would take the full lenght of the GMAT (maybe not). Is there a quick way to solve this problem? Look at the pattern: an = \frac{(n + 2)}{n} So a1 = 3/1 a2 = 4/2 a3 = 5/3 a4 = 6/4 Now when we multiply the first ten terms (n = 1 to 10), \frac{3}{1} * \frac{4}{2} * \frac{5}{3} * \frac{6}{4} * ...* \frac{11}{9} * \frac{12}{10} denominators and numerators will get canceled, 3 with 3, 4 with 4 etc. Only the first two denominators and last two numerators will be left. The product will be 11*12/2 = 66 Since this is a sequence and seeing the pattern " increase of numerator and denominator by 1/1" we determine the next term.This part I understood. But fundamental thing that I have not yet grasped is how to visualise that the respective denominators and numerators actually cancel without enumerating them all. I had to list out all the fractions (just by adding 1/1 to the next) to see that they do cancel and what the final fractions would remain (though in matters of time I did not consume much time-few extra seconds). Is there a logic behind this visualisation ? Please advice. Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 4037 Location: Pune, India Followers: 862 Kudos [?]: 3623 [0], given: 144 Re: Sum of consecutive terms [#permalink] 09 Dec 2010, 11:04 Expert's post helloanupam wrote: VeritasPrepKarishma wrote: Yalephd wrote: The infinite sequence a1, a2,...,an is defined such that an = (n+2) / n for all n ≥ 1. What is the product of the first 10 terms of the sequence? (A) 45 (B) 66 (C) 90 (D) 121 (E) 132 Powering through this would take the full lenght of the GMAT (maybe not). Is there a quick way to solve this problem? Look at the pattern: an = \frac{(n + 2)}{n} So a1 = 3/1 a2 = 4/2 a3 = 5/3 a4 = 6/4 Now when we multiply the first ten terms (n = 1 to 10), \frac{3}{1} * \frac{4}{2} * \frac{5}{3} * \frac{6}{4} * ...* \frac{11}{9} * \frac{12}{10} denominators and numerators will get canceled, 3 with 3, 4 with 4 etc. Only the first two denominators and last two numerators will be left. The product will be 11*12/2 = 66 Since this is a sequence and seeing the pattern " increase of numerator and denominator by 1/1" we determine the next term.This part I understood. But fundamental thing that I have not yet grasped is how to visualise that the respective denominators and numerators actually cancel without enumerating them all. I had to list out all the fractions (just by adding 1/1 to the next) to see that they do cancel and what the final fractions would remain (though in matters of time I did not consume much time-few extra seconds). Is there a logic behind this visualisation ? Please advice. The nth term is (n+2)/n. So I say, when I multiply terms, they will look like this: \frac{[highlight](n+2)[/highlight]}{n} * \frac{(n+3)}{(n+1)} * \frac{(n+4)}{(n+2)} * \frac{(n+5)}{(n+3)} * ... * \frac{(n+12)}{(n+10)} Watch what happens here. The numerator (n+2) of the 1st term has appeared in the denominator of the 3rd term so they will cancel out. So 2nd term will have numerator that is 1 more than the numerator of the 1st term and 4th term will have a denominator that is one more than the denominator of the 3rd term and hence they will be equal and will cancel out. This will continue till we run out of denominators to cancel. So even without enumerating, we know that we will be left with first two numerators and last two denominators. (I wish I could strike the relevant terms with different colored lines but I haven't been able to figure out how to use other operators in fractions!) _________________ Karishma Veritas Prep | GMAT Instructor My Blog Save$100 on Veritas Prep GMAT Courses And Admissions Consulting
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Manager
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Re: Sum of consecutive terms [#permalink]  09 Dec 2010, 20:29
How do we know a1=1? we don't know the value of the first 10 terms? how can we assume a1=1? I didn't see that and was confused by the notation.
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Re: Sum of consecutive terms [#permalink]  09 Dec 2010, 23:56
Expert's post
gettinit wrote:
How do we know a1=1? we don't know the value of the first 10 terms? how can we assume a1=1? I didn't see that and was confused by the notation.

a_1 doesn't equal to 1, it equals to a_1=\frac{1+2}{1}=\frac{3}{1}.

The infinite sequence a1, a2,..., an is defined such that an = (n+2) / n for all n ≥ 1. What is the product of the first 10 terms of the sequence?
(A) 45
(B) 66
(C) 90
(D) 121
(E) 132

Stem gives the formula to get the terms of the sequences: a_n=\frac{n+2}{n} for ALL n\geq{1} (so starting from a_1). So:

a_1=\frac{1+2}{1}=\frac{3}{1};

a_2=\frac{2+2}{2}=\frac{4}{2};

a_3=\frac{3+2}{3}=\frac{5}{3};
...
a_{10}=\frac{10+2}{10}=\frac{12}{10};

The product of the first 10 terms of the sequence will equal: \frac{3}{1}*\frac{4}{2}*\frac{5}{3}*...*\frac{12}{10}=\frac{(3*4*5*6*6*8*9*10)*11*12}{1*2*(3*4*5*6*6*8*9*10)}=\frac{11*12}{2}=66.

Hope it's clear.
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Re: Sum of consecutive terms [#permalink]  10 Dec 2010, 04:39
Thanks Karishma. I took a couple of bounces before I understood from your explanation albeit in own fashion (and hopefully correct!).The way I understand is this- the first two denominators (n & n+1) do not have their mirror numerators and similarly the final two numerators (n+11) & (n+12) wont get to see their mirror denominator images as the process stops at denominator (n+10) and hence dont cancel out.
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Re: Sum of consecutive terms [#permalink]  10 Dec 2010, 04:52
Expert's post
gettinit wrote:
How do we know a1=1? we don't know the value of the first 10 terms? how can we assume a1=1? I didn't see that and was confused by the notation.

This is what is given to you in the question:
The infinite sequence a1, a2,...,an is defined such that an = (n+2) / n for all n ≥ 1.

The product of first 10 terms is asked.

Using the rule, an = (n+2) / n, you can find the nth term easily.
The first term will have n = 1
a1 = (1+2)/1 = 3/1
a2 = (2+2)/2 = 4/2
and so on...
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Karishma
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Save $100 on Veritas Prep GMAT Courses And Admissions Consulting Enroll now. Pay later. Take advantage of Veritas Prep's flexible payment plan options. Veritas Prep Reviews Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 4037 Location: Pune, India Followers: 862 Kudos [?]: 3623 [0], given: 144 Re: Sum of consecutive terms [#permalink] 10 Dec 2010, 04:54 Expert's post helloanupam wrote: Thanks Karishma. I took a couple of bounces before I understood from your explanation albeit in own fashion (and hopefully correct!).The way I understand is this- the first two denominators (n & n+1) do not have their mirror numerators and similarly the final two numerators (n+11) & (n+12) wont get to see their mirror denominator images as the process stops at denominator (n+10) and hence dont cancel out. That's absolutely correct. _________________ Karishma Veritas Prep | GMAT Instructor My Blog Save$100 on Veritas Prep GMAT Courses And Admissions Consulting
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Senior Manager
Joined: 13 Aug 2012
Posts: 465
Concentration: Marketing, Finance
GMAT 1: Q V0
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Kudos [?]: 148 [0], given: 11

The infinite sequence a1, a2,...,an is defined such that an = (n [#permalink]  21 Dec 2012, 05:19
shuj00 wrote:
The infinite sequence a1, a2,...,an is defined such that an = (n+2) / n for all n ≥ 1. What is the product of the first 10 terms of the sequence?

(A) 45
(B) 66
(C) 90
(D) 121
(E) 132

=3*2*\frac{5}{3}*\frac{9}{2}*\frac{1}{2}*\frac{7}{5}*\frac{4}{3}*\frac{9}{7}*\frac{5}{4}*\frac{11}{9}*\frac{12}{10}

Cancel out factors, you are left with 3*2*11=66

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Senior Manager
Joined: 13 Aug 2012
Posts: 465
Concentration: Marketing, Finance
GMAT 1: Q V0
GPA: 3.23
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Kudos [?]: 148 [0], given: 11

The infinite sequence a1, a2,...,an is defined such that an = (n [#permalink]  26 Dec 2012, 18:49
shuj00 wrote:
The infinite sequence a1, a2,...,an is defined such that an = (n+2) / n for all n ≥ 1. What is the product of the first 10 terms of the sequence?

(A) 45
(B) 66
(C) 90
(D) 121
(E) 132

Seeing that the series is a sequence of fractions.
we know that factors will cancel out and easily reveal the answer.

3 * 2 * \frac{5}{3}*\frac{3}{2}*\frac{7}{5}*\frac{8}{6}*\frac{9}{7}*\frac{10}{8}*\frac{11}{9}*\frac{12}{10}

After carefully cancelling out we would be left with

11*6 = 66

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The infinite sequence a1, a2,...,an is defined such that an = (n   [#permalink] 26 Dec 2012, 18:49
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