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705-805 (Hard)|   Sequences|      
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Bunuel
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The sequence of numbers S = {s1, s2, s3 …} is defined by s1 = 2, s2 = 03 Jul 2018, 10:04
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53% (02:03) correct 47% (02:17) wrong based on 410 sessions

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The sequence of numbers S = {s1, s2, s3 …} is defined by \(s_{1} = 2\), \(s_{2} = 10\), and \(S_{n}=(S_{n-1})^{S_{n-2}}\) for each positive integer n greater than or equal to 3. For example, \(s_3 = 10^2\). What is the greatest value of n for which \(s_n\) has 2,000 or fewer digits?

A. 100
B. 20
C. 5
D. 4
E. 3
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D
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The sequence of numbers S = {s1, s2, s3 …} is defined by s1 = 2, s2 = 03 Jul 2018, 10:17
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S4 = 100^10 = 10^20
S5 = (10^20)^100 = 10^2000

S5 will have 2000 zeros and a 1, hence 2001 digits.

Hence n = 4, Option D.
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The sequence of numbers S = {s1, s2, s3 …} is defined by s1 = 2, s2 = 20 Jul 2018, 01:26
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Can any expert explain.
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The sequence of numbers S = {s1, s2, s3 …} is defined by s1 = 2, s2 = 28 Aug 2021, 02:11
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S1 = 2

S2 = 10

Sn = (Sn-1)^(Sn-2)

S3 = 100

S4 = 10^20

S5 = 10^2000 = (2000 zeros and 1 so 2001 digits)

Answer n = 4 D
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GMAT Focus 1: 735 Q90 V89 DI81
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The sequence of numbers S = {s1, s2, s3 …} is defined by s1 = 2, s2 = 28 Aug 2021, 03:16
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Bunuel
The sequence of numbers S = {s1, s2, s3 …} is defined by \(s_{1} = 2\), \(s_{2} = 10\), and \(S_{n}=(S_{n-1})^{S_{n-2}}\) for each positive integer n greater than or equal to 3. For example, \(s_3 = 10^2\). What is the greatest value of n for which \(s_n\) has 2,000 or fewer digits?

A. 100
B. 20
C. 5
D. 4
E. 3
Show more


\(s_{1} = 2\),
\(s_{2} = 10\), and
\(S_{n}=(S_{n-1})^{S_{n-2}}\)
\(s_3 = 10^2=100\)
\(s_4 = 100^{10}=10^{20}\)

So we can say that the values are of the type 100000….
Any number of this type has number of digits=number of zeroes+1

For example: 10^2 will have 2+1
10^200 will have 200+1.

For a number to have 2000 or less digits, the power should be 1999 or less.

\(s_5 =(10^{20})^{100}=10^{2000}\), which will have 2000+1 or 2001 digits.

Thus, the answer is 4.

D
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The sequence of numbers S = {s1, s2, s3 …} is defined by s1 = 2, s2 = 12 Apr 2025, 12:11
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The mistake we commonly make in such questions is that:

To determine S4, Instead of capturing 10^2^10, We might end up taking 10^2^2, because we might consider Sn-2 as n-2 and won't be taking value of Sn-2.

Bunuel
The sequence of numbers S = {s1, s2, s3 ...} is defined by \(s_{1} = 2\), \(s_{2} = 10\), and \(S_{n}=(S_{n-1})^{S_{n-2}}\) for each positive integer n greater than or equal to 3. For example, \(s_3 = 10^2\). What is the greatest value of n for which \(s_n\) has 2,000 or fewer digits?

A. 100
B. 20
C. 5
D. 4
E. 3
Show more
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Bunuel
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