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25 Aug 2014, 12:01
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B
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
45%
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Question Stats:
69% (02:00) correct
31%
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In the sequence a1,a2,a3...an, each term an is defined as an=1/(−n)^n. The sum of the first ten terms of the sequence (a1,a2,a3...a10) must be __________.
A) Less than −1
B) Between −1 and −1/2
C) Between −1/2 and 0
D) Between 0 and 1/2
E) Greater than 1/2
A) Less than −1
B) Between −1 and −1/2
C) Between −1/2 and 0
D) Between 0 and 1/2
E) Greater than 1/2
09 Mar 2016, 02:44
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oanhnguyen1116
Hi,
Since there has been some confusion till now on this Q, let me give you a method
one approach that can be useful is--
Quote:
Solution--
\(a_n=\frac{1}{{-n}^n}\)
so
1) \(a_1=\frac{1}{{-1}^1}\)
2) \(a_2=\frac{1}{{-2}^2}\)..
3) \(a_3=\frac{1}{{-3}^3}\)...
4) \(a_4=\frac{1}{{-4}^4}\)..
INFERENCE:-
1)Starting from -1, alternate terms are -ive and positive..
2) Numeric value keeps decreasing as we go up..
so two scenarios--
1) (first term + second term) + (third term+fourth term) +... (ninth+tenth)..
=> \((-1+\frac{1}{4}) + (-\frac{1}{3^3} + \frac{1}{4^4}) ....+ (-\frac{1}{9^9} + \frac{1}{10^10})\)..
\(-\frac{3}{4} + (- ..) + (- ..)=- \frac{3}{4} - .. - ......\)
keeping the inference in mind, each pair will result in a negative qty
so the SUM will be lesser than \(-\frac{3}{4}\)..
2) lets take first term and then pair of 2nd-3rd, 4th-5th.. and so on
first term + (second term+third term)+(fourth term+fifth term)...(eighth+ninth)+tenth..
=>\(-1+ (\frac{1}{4}-\frac{1}{3^3} ) + ( \frac{1}{4^4}-..) .... (\frac{1}{8^8}-\frac{1}{9^9}) + \frac{1}{10^10}\)..
so leaving -1 and \(\frac{1}{10^10}\), all pair will result in a positive value..
so\(-1+\frac{1}{10^10} +\) some positive value + some positive value +...
so SUM has to be >-1..
Now we have two limits for the SUM..
\(SUM<-\frac{3}{4}\)and\(Sum>-1\)..
so it lies between -1 and -3/4..
choice B contains this RANGE..
so ans B. Between −1 and −1/2
General Discussion
25 Aug 2014, 12:05
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Can anyone explain better than the solution below?
As with most sequence problems, the biggest step to take is to jot out the first few terms in search of a pattern or some other insight about the way the sequence unfolds. Here, that gives you:
a1=1/(−1)^1=−1
a2=1/(−2)^2=1/4
a3=1/(−3)^3=−1/27
a4=1/(−4)^4=1/256
What you should see here is that the odd terms are all negative and the evens are positive, and that the terms get so small so quickly as to be inconsequential. After just adding the first few terms you should see that you have a little less than −3/4 and that the remaining terms are going to be too small to swing that value very much in either direction, so the answer must be B.
As with most sequence problems, the biggest step to take is to jot out the first few terms in search of a pattern or some other insight about the way the sequence unfolds. Here, that gives you:
a1=1/(−1)^1=−1
a2=1/(−2)^2=1/4
a3=1/(−3)^3=−1/27
a4=1/(−4)^4=1/256
What you should see here is that the odd terms are all negative and the evens are positive, and that the terms get so small so quickly as to be inconsequential. After just adding the first few terms you should see that you have a little less than −3/4 and that the remaining terms are going to be too small to swing that value very much in either direction, so the answer must be B.
