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Re: The sequence a1, a2, a3, ..., an, ... is such that an=1/n^21/(n1)^2 [#permalink]
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Bunuel wrote:
The sequence \(a_1, \ a_2, \ a_3, \ ..., \ a_n, \ ...\) is such that \(a_n=\frac{1}{n^2}−\frac{1}{(n−1)^2}\) for all integers \(n≥2\) and \(a_1 = 1\). Is the sum of this sequence greater than 0.005?

(1) The number of terms in the sequence is greater than 12

(2) The number of terms in the sequence is lesser than 14


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\(a_n= \frac{1}{n^2}−\frac{1}{(n−1)^2}\)...
\(a_1=1\)

\(a_2= \frac{1}{2^2}−\frac{1}{(2−1)^2}\)...
The sum is \(a_1+a_2=1+ \frac{1}{2^2}−\frac{1}{(2−1)^2}=\frac{1}{2^2}\)

\(a_3= \frac{1}{3^2}−\frac{1}{(3−1)^2}\)...
The sum is \(a_1+a_2+a_3=\frac{1}{2^2}+\frac{1}{3^2}−\frac{1}{(3−1)^2}=\frac{1}{3^2}\)...

So, the sum is \(\frac{1}{n^2}\), where n is the number of terms.

Now, \(0.005=\frac{5}{1000}=\frac{1}{200}\) and \(\frac{1}{14^2}>\frac{1}{200}>\frac{1}{15^2}\)
So, we are checking whether the number of terms is less than 15 OR greater than or equal to 15.

(1) The number of terms in the sequence is greater than 12
So, n could be 13 or 14, then the answer would be YES. But 15 and above would give answer as NO.
Insufficient

(2) The number of terms in the sequence is lesser than 14
So, n is always less than 15, and the answer would be YES always.
Sufficient


B

Trent981, I believe OA has been marked C erroneously.
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Re: The sequence a1, a2, a3, ..., an, ... is such that an=1/n^21/(n1)^2 [#permalink]
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Re: The sequence a1, a2, a3, ..., an, ... is such that an=1/n^21/(n1)^2 [#permalink]
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