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The sequence S is defined by Sn = Sn – 1 + Sn – 2 – 1 for each integer

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Bunuel
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The sequence S is defined by Sn = Sn – 1 + Sn – 2 – 1 for each integer [#permalink] New post   27 Jul 2018, 01:23
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The sequence S is defined by \(S_n = S_{n - 1} + S_{n - 2} - 1\) for each integer n ≥ 3. If \(S_1 = 11\) and \(S_3 = 10\), what is the value of \(S_5\)?

(A) 0
(B) 9
(C) 10
(D) 18
(E) 19
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D
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Re: The sequence S is defined by Sn = Sn – 1 + Sn – 2 – 1 for each integer [#permalink] New post   27 Jul 2018, 02:00
Sn = Sn−1 + Sn−2 − 1
S1=11 and S3=10, S5 = ?

S3 = S2 + S1 − 1
10 = S2 + 11 - 1
S2 = 0

S4 = S3 + S2 − 1
S4 = 10 + 0 - 1 = 9

S5 = S4 + S3 − 1
S5 = 9 + 11 - 1 = 19

Hence, E.
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Re: The sequence S is defined by Sn = Sn – 1 + Sn – 2 – 1 for each integer [#permalink] New post   28 Aug 2018, 07:49
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10=S2+11-1,S2=0

S5=(S4)+S3-1=(S3+S2-1)+S3-1=10+10+0-1-1=18
Answer D.
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Re: The sequence S is defined by Sn = Sn – 1 + Sn – 2 – 1 for each integer [#permalink] New post   25 Oct 2018, 15:59
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Bunuel wrote:
The sequence S is defined by \(S_n = S_{n - 1} + S{n - 2} - 1\) for each integer n ≥ 3. If \(S_1 = 11\) and \(S_3 = 10\), what is the value of \(S_5\)?

(A) 0
(B) 9
(C) 10
(D) 18
(E) 19


Bunuel Could you please format the equation properly? It's currently confusing. Thanks!

\(S_n = S_{n - 1} + S{n - 2} - 1\) —> \(S_n = S_{n - 1} + S_{n - 2} - 1\)
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Re: The sequence S is defined by Sn = Sn – 1 + Sn – 2 – 1 for each integer [#permalink] New post   25 Oct 2018, 21:12
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dabaobao wrote:
Bunuel wrote:
The sequence S is defined by \(S_n = S_{n - 1} + S{n - 2} - 1\) for each integer n ≥ 3. If \(S_1 = 11\) and \(S_3 = 10\), what is the value of \(S_5\)?

(A) 0
(B) 9
(C) 10
(D) 18
(E) 19


Bunuel Could you please format the equation properly? It's currently confusing. Thanks!

\(S_n = S_{n - 1} + S{n - 2} - 1\) —> \(S_n = S_{n - 1} + S_{n - 2} - 1\)

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Edited. Thank you.
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Re: The sequence S is defined by Sn = Sn – 1 + Sn – 2 – 1 for each integer [#permalink] New post   26 Oct 2018, 17:24
Strategy: Utilize a Table for the sequence to help understand how the sequence works

Given
s1 = 11
s3 = 10 = s2 + s1 - 1

Question
s5 = ?

Equations used
sn = s(n-1) + s(n-2) + 1

Calculations
s5 = s4 + s3 - 1
s5 = s4 + 11 - 1
What is s4?

s4 = s3 + s2 - 1
s4 = 11 + s2 - 1
To find s4, we need to know what s2 is

From the Given:
s3 = s2 + s1 - 1
s3 - s1 + 1 = s2
10 - 11 + 1 = s2
0 = s2


s4 = s3 + s2 -1
s4 = 10 + 0 - 1
s4 = 9

s5 = s4 + s3 - 1
s5 = 9 + 10 - 1
s5 = 18
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Re: The sequence S is defined by Sn = Sn – 1 + Sn – 2 – 1 for each integer [#permalink] New post   26 Oct 2018, 18:43
Bunuel wrote:
The sequence S is defined by \(S_n = S_{n - 1} + S_{n - 2} - 1\) for each integer n ≥ 3. If \(S_1 = 11\) and \(S_3 = 10\), what is the value of \(S_5\)?

(A) 0
(B) 9
(C) 10
(D) 18
(E) 19


Steps:
1) Compute S_2 = S_3 - S_1 + 1 = 0
2) Using the same method, compute S_4 = 10 - 1 = 9
3)Now, compute S_5 = 9 + 10 - 1 = 18

Correct answer: D

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The sequence S is defined by Sn = Sn – 1 + Sn – 2 – 1 for each integer [#permalink] New post   21 Jan 2019, 21:56
Good night Bunuel !

I am trying to understand this problem but still do not get it, why are we assuming that S1 + S2 will be equal to S3?

Does this mean that every integer after the last one is going to be -1 from the sum of the las two? I think my main problem is that I am not understanding the wording well enough.

\(S_n = S_{n - 1} + S_{n - 2} - 1\)

Could someone help me, please?

Kind regards!
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Bunuel
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Re: The sequence S is defined by Sn = Sn – 1 + Sn – 2 – 1 for each integer [#permalink] New post   21 Jan 2019, 23:04
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jfranciscocuencag wrote:
Good night Bunuel !

I am trying to understand this problem but still do not get it, why are we assuming that S1 + S2 will be equal to S3?

Does this mean that every integer after the last one is going to be -1 from the sum of the las two? I think my main problem is that I am not understanding the wording well enough.

\(S_n = S_{n - 1} + S_{n - 2} - 1\)

Could someone help me, please?

Kind regards!


We are told that the sequence S is defined by \(S_n = S_{n - 1} + S_{n - 2} - 1\) for each integer n ≥ 3.

This means that \(S_n = S_{n - 1} + S_{n - 2} - 1\) is a formula which gives you every term of the sequence starting from 3rd. Look at the formula, it says that \(S_n\) equals to the sum of two preceding terms, \(S_{n - 1}\) and \(S_{n - 2}\), minus 1. For example:

\(S_3 = S_{3 - 1} + S_{3 - 2} - 1=S_{2} + S_{1} - 1\);
\(S_4 = S_{4 - 1} + S_{4 - 2} - 1=S_{3} + S_{2} - 1\);
\(S_5 = S_{5 - 1} + S_{5 - 2} - 1=S_{4} + S_{3} - 1\);
...

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Re: The sequence S is defined by Sn = Sn – 1 + Sn – 2 – 1 for each integer [#permalink]
21 Jan 2019, 23:04
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