fitzpratik wrote:
In a sequence S, each term, after first term, is obtained by multiplying the previous term by a constant k, where k ≠ 1. How many terms in the sequence are less than 1?
A. 8th term in the sequence is 32
B. 2nd term in the sequence is \(\frac{1}{2}\)
Solution
Step 1: Analyse Question Stem
• Let a be the first term of the sequence.
o Second term = \(k*a\)
o Third term = \(k^2*a\)
o nth term = \(k^{(n-1)}*a\)
We need to find the number of terms less than 1.
Step 2: Analyse Statements Independently (And eliminate options) – AD/BCE
Statement 1: \(8th\ term = 32\)
• \(k^{(8-1)}*a=32\)
• \(k^7*a = 32\)
o We cannot infer anything about the value of terms.
Hence, statement 1 is not sufficient, we can eliminate answer options A and D.
Statement 2: \(2th\ term = \frac{1}{2}\)
• \(k^{(2-1)}*a = \frac{1}{2} \)
• \(k*a = \frac{1}{2}\\
\)o We cannot infer anything about the value of terms.
Hence, statement 2 is also not sufficient we can eliminate answer options B.
Step 3: Analyse Statements by combining.
From statement 1:\( k^7*a = 32\)…(i)
From statement 2: \(k*a = \frac{1}{2}\) …(ii)
On dividing equation (i) by (ii), we get
\(\frac{k^7*a}{k*a} =\frac{32}{\frac{1}{2}}\)
\(k^6 =64\)
Now, there are two possible values of \(k, 2 and -2\).
• If \(k = 2\), then \(a = \frac{1}{4}\)
o Number of terms less than 1 will be 2 only
• If \(k= -2\), then \(a = -\frac{1}{4}\)
o Every alternate term will be negative, which will be less than 1, we can’t find the number of terms less than 1.
Hence, the correct answer is
Option E.
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