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15 Feb 2023, 21:36
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C
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Difficulty:
95%
(hard)
Question Stats:
35% (02:09) correct
65%
(02:19)
wrong
based on 92
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In a sequence S, each term, after the first term, is obtained by multiplying the previous term by a constant k, where k is an integer other than 1. How many terms in the sequence are less than 1?
(1) The 8th term in the sequence is 32 and the 2nd term in the sequence is \(\frac{1}{2}\)
(2) The fifth term in the sequence is 4.
(1) The 8th term in the sequence is 32 and the 2nd term in the sequence is \(\frac{1}{2}\)
(2) The fifth term in the sequence is 4.
15 Feb 2023, 23:48
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TBT
Assume that the first term in the sequence is x
Hence, the sequence can be represented as
\(x, \quad xk, \quad xk^2, \quad x^3, \quad ...... \quad x^{n-1}\)
\(n^{\text{th}} \text{term of the sequence}= x^{n-1}\)
Statement 1
(1) The 8th term in the sequence is 32 and the 2nd term in the sequence is \(\frac{1}{2}\)
8th term = \(xk^7\) = 32
2nd term = \(xk = \frac{1}{2}\)
\(\frac{\text{8th term} }{ \text{2nd term}} = k^6 = 64\)
\(k = \pm 2\)
k = +2; x = 1/4
k = -2; x = -1/4
As we have two combinations, we cannot predict many terms in the sequence are less than 1.
Statement 2
\(xk^4 = 4\)
Clearly insufficient.
Eliminate B
Combined
As \(xk^4 = 4\)
\(k^4\) is positive, therefore for the multiplication to be a positive term x has to be positive.
Therefore x = 1/4 & k = 2
Once, we have the values in place, we can formulate the sequence and count the number of terms that are less than 1.
Sufficient.
Option C
TBT
Question: Between \(a_1\) and \(a_8\), No of terms < 1?
Given sequence is a GP: \(a, ar_1, ar_2, .......\)
where \(a_n=ar^{n-1}\)
(1) The 8th term in the sequence is 32 and the 2nd term in the sequence is \(\frac{1}{2}\)
\(a_8=ar^7=32\)
\(a_2=ar^1=\frac{1}{2}\)
\(\frac{{ar^7}}{{ar^1}}=r^6=64\)
\(r=\pm 2\)
\(1/2,1,2,4,...... (1)\)
\(-1/2,1,-2,4,...... (2)\) NOT SUFFICIENT
(2) The fifth term in the sequence is 4.
\(a_4=ar^3=4\) NOT SUFFICIENT
Combining:
Hence, only scenario (1) is possible SUFFICIENT (C)
