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18 Mar 2020, 09:19
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D
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
65%
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Question Stats:
58% (02:09) correct
42%
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wrong
based on 225
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The \(n_{th}\) term of a sequence of distinct positive integers is determined by \(a_n = k^{a_{n-2}}*a_{n-1}\), where n ≥ 3 and k is a constant. If \(a_2 = 4\), and \(a_3 = 36\), what is the value of \(a_5\) ?
(1) \(a_1 = 2\)
(2) \(k = 3\)
Are You Up For the Challenge: 700 Level Questions
(1) \(a_1 = 2\)
(2) \(k = 3\)
Are You Up For the Challenge: 700 Level Questions
18 Mar 2020, 13:11
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Bunuel
Solution
Step 1: Analyse Question Stem
- • Given sequence is a sequence of distinct positive integers.
• nth term of the sequence \(= a_n = k^{a_{n-2}}*a_{n-1}……..Eq(i)\)
- o for all n≥3
o k is constant
• \(a_3 = 36\)
• We need to find the value of \(a_5\)
- o From Eq.(i), we can write, \(a_5 = k^{a_3}*a_4 ……..Eq.(ii)\)
o Now, \(a_4 = k^{a_2}*a_3 \)
- Substituting the value of \(a_2\) and \(a_3\) in the above equation, we get
\(a_4 = k^4*36\)
- \(a_5 = k^{36}*k^4*36 =k^{40}*36 ………Eq(iii)\)
Step 2: Analyse Statements Independently (And eliminate options) – AD/BCE
Statement 1: \(a_1 = 2\)
- • From Eq(i) we can write, \(a_3 =k^{a_1}*a_2 = k^{2}*a_2\)
• Now, substituting the value of \(a_2\) and \(a_3\) in the above equation, we get
- o \(36 =k^2*4⟹k^2 = 9\)
o \((k^2)^{20} = 9^{20}\)
o Or, \(k^{40} = 9^{20}\)
Hence, statement 1 is sufficient and we can eliminate answer Options B, C and E.
Statement 2: \(k=3\)
• From this statement we got the value of k. So we can easily substitute it in Eq.(iii) and find the value of \(a_5\).
Hence, statement 2 is also sufficient and the correct answer is Option D.
08 Aug 2020, 07:08
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I'm confused by this question because the sequence is defined for n ≥ 3 and yet \(a_1\) and \(a_2\) apparently have values. I would have thought we can't know the values of \(a_1\) and \(a_2\) because the sequence is only defined for n ≥ 3. I understand how to find the answer assuming that what we're given is accurate.
