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# Sequence S is defined as Sn=Sn-1 + 1 +1/(Sn-1 + 1) for all n

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Sequence S is defined as Sn=Sn-1 + 1 +1/(Sn-1 + 1) for all n [#permalink]

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19 Dec 2012, 07:18
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Sequence S is defined as Sn = (Sn-1 +1) + {1 / (Sn-1 +1)} for all n > 1. If S1 = 100, then which of the following must be true of Q, the sum of the first 16 terms of S?

(A) 1,600 ≤ Q ≤ 1,650
(B) 1,650 ≤ Q ≤ 1,700
(C) 1,700 ≤ Q ≤ 1,750
(D) 1,750 ≤ Q ≤ 1,800
(E) 1,800 ≤ Q ≤ 1,850
[Reveal] Spoiler: OA

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Last edited by daviesj on 20 Dec 2012, 00:09, edited 1 time in total.

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Re: Sequence S is defined as for all n > 121. [#permalink]

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19 Dec 2012, 08:49
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daviesj wrote:
Sequence S is defined as Sn = (Sn-1 +1) + {1 / (Sn-1 +1)} for all n > 1. If S1 = 100, then which of the following must be true of Q, the sum of the first 16 terms of S?
(A) 1,600 ≤ Q ≤ 1,650
(B) 1,650 ≤ Q ≤ 1,700
(C) 1,700 ≤ Q ≤ 1,750
(D) 1,750 ≤ Q ≤ 1,800
(E) 1,800 ≤ Q ≤ 1,850

method to solve plz...

$$S_1 = 100$$

$$S_2 = \frac{101^2 + 1}{101} \approx 101$$ (Since 1 is negligible when compared to $$101^2$$)

So, the series is almost an arithmetic progression with a=100, d=1,

We have got to find the sum of "n" terms where "n" is 16.

$$S_{16} = \frac{16}{2}*(2*100 + (16-1)*1)$$

= 8*215 = 1720

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Re: Sequence S is defined as for all n > 121. [#permalink]

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19 Dec 2012, 21:14
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daviesj wrote:
Sequence S is defined as Sn = (Sn-1 +1) + {1 / (Sn-1 +1)} for all n > 1. If S1 = 100, then which of the following must be true of Q, the sum of the first 16 terms of S?
(A) 1,600 ≤ Q ≤ 1,650
(B) 1,650 ≤ Q ≤ 1,700
(C) 1,700 ≤ Q ≤ 1,750
(D) 1,750 ≤ Q ≤ 1,800
(E) 1,800 ≤ Q ≤ 1,850

method to solve plz...

S1 has been given a big value i.e. 100 instead of the usual 0/1 etc. Why? Because 1/100 is negligible when added to 101

$$S_n = S_{n-1} + 1 + \frac{1}{S_{n-1}}$$

$$S_2 = S_{1} + 1 + \frac{1}{S_{1}} = 100 + 1 + \frac{1}{100} = 101$$ approx

$$S_3 = 101+ 1 + 1/101 = 102$$ approx
.
.
$$S_{16} = 115$$approx

$$S_1 + S_2 + ...S_{16} = 100 + 101 + 102 + ... 115 = 16*100 + 15*16/2 = 1720$$

The sum will be a little more than 1720.
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Re: Sequence S is defined as for all n > 121. [#permalink]

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19 Dec 2012, 22:07
VeritasPrepKarishma wrote:
daviesj wrote:
Sequence S is defined as Sn = (Sn-1 +1) + {1 / (Sn-1 +1)} for all n > 1. If S1 = 100, then which of the following must be true of Q, the sum of the first 16 terms of S?
(A) 1,600 ≤ Q ≤ 1,650
(B) 1,650 ≤ Q ≤ 1,700
(C) 1,700 ≤ Q ≤ 1,750
(D) 1,750 ≤ Q ≤ 1,800
(E) 1,800 ≤ Q ≤ 1,850

method to solve plz...

S1 has been given a big value i.e. 100 instead of the usual 0/1 etc. Why? Because 1/100 is negligible when added to 101

$$S_n = S_{n-1} + 1 + \frac{1}{S_{n-1}}$$

$$S_2 = S_{1} + 1 + \frac{1}{S_{1}} = 100 + 1 + \frac{1}{100} = 101$$ approx

$$S_3 = 101+ 1 + 1/101 = 102$$ approx
.
.
$$S_{16} = 115$$approx

$$S_1 + S_2 + ...S_{16} = 100 + 101 + 102 + ... 115 = 16*100 + 15*16/2 = 1720$$

The sum will be a little more than 1720.

i would say the question is wrong or, atleast, not an exact GMAT type question...why to assume N as an integer...it is not specified in the question that n is an integer....n could be 1.2, 1.2,....etc for n>1 when n is not an integer.... i am thinking in the GMAT prospective...what is the source of this qtn?

Last edited by muralilawson on 19 Dec 2012, 22:53, edited 2 times in total.

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Re: Sequence S is defined as for all n > 121. [#permalink]

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19 Dec 2012, 22:19
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daviesj wrote:
Sequence S is defined as Sn = (Sn-1 +1) + {1 / (Sn-1 +1)} for all n > 1. If S1 = 100, then which of the following must be true of Q, the sum of the first 16 terms of S?
(A) 1,600 ≤ Q ≤ 1,650
(B) 1,650 ≤ Q ≤ 1,700
(C) 1,700 ≤ Q ≤ 1,750
(D) 1,750 ≤ Q ≤ 1,800
(E) 1,800 ≤ Q ≤ 1,850

method to solve plz...

S1 = $$100$$

S2 = $$100 + (1 + \frac{1}{101})$$
If you will notice $$1 + \frac{1}{101}$$ is approximately 1... S2 = 100 + 1 is approx. ~ 101

S3 = $$101 + (1 + \frac{1}{101})$$
If you wil notice $$1 + \frac{1}{101}$$ is approximately 1... S3 = 101 + 1 approx. ~ 102

Sum = 16 * 100 + 1 + 2 + 3 + ... + 15 = $$1600 + \frac{15(15+1)}{2} = 1600 + 120 = 1720$$

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Re: Sequence S is defined as for all n > 121. [#permalink]

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19 Dec 2012, 22:52
muralilawson wrote:
VeritasPrepKarishma wrote:
daviesj wrote:
Sequence S is defined as Sn = (Sn-1 +1) + {1 / (Sn-1 +1)} for all n > 1. If S1 = 100, then which of the following must be true of Q, the sum of the first 16 terms of S?
(A) 1,600 ≤ Q ≤ 1,650
(B) 1,650 ≤ Q ≤ 1,700
(C) 1,700 ≤ Q ≤ 1,750
(D) 1,750 ≤ Q ≤ 1,800
(E) 1,800 ≤ Q ≤ 1,850

method to solve plz...

S1 has been given a big value i.e. 100 instead of the usual 0/1 etc. Why? Because 1/100 is negligible when added to 101

$$S_n = S_{n-1} + 1 + \frac{1}{S_{n-1}}$$

$$S_2 = S_{1} + 1 + \frac{1}{S_{1}} = 100 + 1 + \frac{1}{100} = 101$$ approx

$$S_3 = 101+ 1 + 1/101 = 102$$ approx
.
.
$$S_{16} = 115$$approx

$$S_1 + S_2 + ...S_{16} = 100 + 101 + 102 + ... 115 = 16*100 + 15*16/2 = 1720$$

The sum will be a little more than 1720.

i would say the question is wrong or, atleast, not an exact GMAT question...why to assume N as an integer...it is not specified in the question that n is an integer....n could be 1.2, 1.2,....etc for n>1 when n is not an integer.... i am thinking in the GMAT prospective...

Since this is a PS question and not a DS question, we are free to make that assumption. "n" is only a subscript indicating the ordinal number of each term and hence can be taken to be integers.
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Re: Sequence S is defined as Sn=Sn-1 + 1 +1/(Sn-1 + 1) for all n [#permalink]

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Re: Sequence S is defined as Sn=Sn-1 + 1 +1/(Sn-1 + 1) for all n [#permalink]

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Re: Sequence S is defined as Sn=Sn-1 + 1 +1/(Sn-1 + 1) for all n [#permalink]

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10 Nov 2016, 00:25
daviesj wrote:
Sequence S is defined as Sn = (Sn-1 +1) + {1 / (Sn-1 +1)} for all n > 1. If S1 = 100, then which of the following must be true of Q, the sum of the first 16 terms of S?

(A) 1,600 ≤ Q ≤ 1,650
(B) 1,650 ≤ Q ≤ 1,700
(C) 1,700 ≤ Q ≤ 1,750
(D) 1,750 ≤ Q ≤ 1,800
(E) 1,800 ≤ Q ≤ 1,850

Please check the solution in attachment
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Re: Sequence S is defined as Sn=Sn-1 + 1 +1/(Sn-1 + 1) for all n   [#permalink] 10 Nov 2016, 00:25
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