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# For every positive integer n, the nth term of sequence is given by an=

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For every positive integer n, the nth term of sequence is given by an= [#permalink]

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24 Apr 2016, 10:38
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For every positive integer n, the nth term of sequence is given by an= 1/n - 1/(n+1). What is the sum of the first 100 terms?

(a) 1
(b) 0
(c) 25
(d) 99/100
(e) 100/101
[Reveal] Spoiler: OA

Last edited by Bunuel on 24 Apr 2016, 11:26, edited 1 time in total.
Edited the question.
Math Expert
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For every positive integer n, the nth term of sequence is given by an= [#permalink]

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24 Apr 2016, 10:47
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inakihernandez wrote:
For every positive integer n, the nth term of sequence is given by an= 1/n - 1/n+1. What is the sum of the first 100 terms?

(a) 1
(b) 0
(c) 25
(d) 99/100
(e) 100/101

Hi,
the moment you see the formula of nth number, it should tell you we are looking at a sequence ..
$$a_1 = \frac{1}{1}-\frac{1}{2}$$..
$$a_2= \frac{1}{2}-\frac{1}{3}$$..
$$a_3 = \frac{1}{3}-\frac{1}{4}$$..
.....
$$a_{99} = \frac{1}{99}-\frac{1}{100}$$..
$$a_{100} = \frac{1}{100}- \frac{1}{101}$$..
ADD all
$$1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}..... \frac{1}{99}-\frac{1}{100}+\frac{1}{100}-\frac{1}{101} = 1-\frac{1}{101} = \frac{100}{101}$$
so when you add all terms, what is left is $$1-\frac{1}{101} = \frac{100}{101}$$..
E
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Re: For every positive integer n, the nth term of sequence is given by an= [#permalink]

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24 Apr 2016, 11:21
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Hi chetan2u! thank you very much for your answer.

There is any way to calculate it using the formula (first term+last term)/2*number of terms?

because what I did as a first approach was a1= 1/2 ; a100= almost 0 => (1/2+0)/2*100= 25

I understood your solution but I was wondering if there is any way using the formula below, or was I using a wrong approach?
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Re: For every positive integer n, the nth term of sequence is given by an= [#permalink]

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24 Apr 2016, 11:26
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inakihernandez wrote:
Hi chetan2u! thank you very much for your answer.

There is any way to calculate it using the formula (first term+last term)/2*number of terms?

because what I did as a first approach was a1= 1/2 ; a100= almost 0 => (1/2+0)/2*100= 25

No, don't use this formula here..
use it ONLY in an AP, where difference is common in consecutive numbers..
If you want to do approx, then too the terms should be atleast close to be called in AP
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Re: For every positive integer n, the nth term of sequence is given by an= [#permalink]

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04 May 2016, 00:30
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inakihernandez wrote:
For every positive integer n, the nth term of sequence is given by an= 1/n - 1/(n+1). What is the sum of the first 100 terms?

(a) 1
(b) 0
(c) 25
(d) 99/100
(e) 100/101

For similar questions, check:
http://www.veritasprep.com/blog/2012/03 ... sequences/
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Re: gmat prep Q DS (series) [#permalink]

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08 May 2016, 22:27
This clearly look like a GP series

a1 = 1 - 1/2 = 1/2

a2 = 1/2 - 1/3 = 1/6

r = 1/3

S100 = 1/2 (( 1 - (1/3)^100)/(1 - 1/3))

= 3/4 * (1 - 1/3)^100)

------> is this correct? how to solve beyond this to reach the ans?
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For every positive integer n, the nth term of sequence is given by an= [#permalink]

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08 May 2016, 22:54
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aniketm.87@gmail.com wrote:
For every positive integer n, the nth term of a sequence is given by an = 1/n - 1(n + 1). what is the sum of the 1st 100 terms in this sequence?

A) 0

B) 1/101

C) 99/100

D) 100/101

E) 1

hi
its not Gp..
At times we complicate the Q while simplifying it...

first write the numbers..
1st = $$1-\frac{1}{2}$$..
2nd = $$\frac{1}{2}-\frac{1}{3}$$
3rd = $$\frac{1}{3}-\frac{1}{4}$$..
99th = $$\frac{1}{99}-\frac{1}{100}$$..
1ooth = $$\frac{1}{100}-\frac{1}{101}$$...
add all ..
$$1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}..... \frac{1}{99}-\frac{1}{100}+\frac{1}{100}-\frac{1}{101} = 1-\frac{1}{101} = \frac{100}{101}$$
D
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Re: For every positive integer n, the nth term of sequence is given by an= [#permalink]

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14 May 2016, 14:43
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Bunuel, chetan2u,

For me, the tricky part here is that the GMAT assumes that we know that (1- 1/2) + (1/2 - 1/4) + (1/4 - 1/8) ... (1/99-1/100)+(1/100-1/101) = 1
How can we know that? Is there any specific rule?

Many thanks in advance,

Jaime
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For every positive integer n, the nth term of sequence is given by an= [#permalink]

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14 May 2016, 20:09
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Jaikuz wrote:
Bunuel, chetan2u,

For me, the tricky part here is that the GMAT assumes that we know that (1- 1/2) + (1/2 - 1/4) + (1/4 - 1/8) ... (1/99-1/100)+(1/100-1/101) = 1
How can we know that? Is there any specific rule?

Many thanks in advance,

Jaime

Hi,

Except the first and the last term, all the other terms get cancelled out. So the only calculation required here is 1 - 1/101.
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Re: For every positive integer n, the nth term of sequence is given by an= [#permalink]

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15 May 2016, 11:31
Hi Vyshak,

Can you please explain in more detail? I still don't see it.

Many thanks in advance.
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Re: For every positive integer n, the nth term of sequence is given by an= [#permalink]

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15 May 2016, 12:13
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Jaikuz wrote:
Hi Vyshak,

Can you please explain in more detail? I still don't see it.

Many thanks in advance.

an= 1/n - 1/(n+1).

a1 = 1 - 1/2
a2 = 1/2 - 1/3
a3 = 1/3 - 1/4
.
.
.
a99 = 1/99 - 1/100
a100 = 1/100 - 1/101

Question: a1 + a2 + a3 + ...... + a99 + a100
= 1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + ......................... + 1/99 - 1/100 + 1/100 - 1/101 = 1 - 1/101 = 100/101

As you can see, except the first and the last terms the remaining terms get cancelled out.

Hope it helps.
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Re: For every positive integer n, the nth term of sequence is given by an= [#permalink]

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30 Oct 2016, 10:25
Vyshak chetan2u : Why can't we use the Sum formula in this case? S100 = 100/2 * (2a + (100-1)d) ?

'Cuz if we do use, the answer comes to a value too far out from the answer choices, as per calculations.

Am i missing something?
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Re: For every positive integer n, the nth term of sequence is given by an= [#permalink]

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30 Oct 2016, 10:41
Vishvesh88 wrote:
Vyshak chetan2u : Why can't we use the Sum formula in this case? S100 = 100/2 * (2a + (100-1)d) ?

'Cuz if we do use, the answer comes to a value too far out from the answer choices, as per calculations.

Am i missing something?

The sum formula that you have mentioned can be applied only when the sequence is in arithmetic progression. The given sequence is not in AP and hence cannot be applied.
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Re: For every positive integer n, the nth term of sequence is given by an= [#permalink]

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17 Dec 2017, 12:00
Hello,

Adding information regarding why this is not an AP (arithmetic progression) nor a GP (geometric progression). Do the math for:
a1 = 1/2
a2 = 1/6
a3 = 1/12
a4 = 1/20
a5 = 1/30

If you try to put in any general term formula:
a2 = a1 + r.1 => r = -1/3
Trying to apply for a3: a3 = 1/12 <> 1/2 + 2.(-1/3)
The same goes with GP.

Best,
Re: For every positive integer n, the nth term of sequence is given by an=   [#permalink] 17 Dec 2017, 12:00
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