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23 Oct 2018, 22:03
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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:
55%
(hard)
Question Stats:
70% (02:43) correct
30%
(03:03)
wrong
based on 396
sessions
History
Date
Time
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Not Attempted Yet
A series is defined, for positive integer n, by
\(a_n=\frac{n+1}{n+3}−\frac{n}{n+2}\)
What is the sum of the first 60 terms of this series?
A. 1/6
B. 1/3
C. 29/30
D. 40/63
E. 61/63
\(a_n=\frac{n+1}{n+3}−\frac{n}{n+2}\)
What is the sum of the first 60 terms of this series?
A. 1/6
B. 1/3
C. 29/30
D. 40/63
E. 61/63
eswarchethu135
GMAT 2: 640 Q49 V27
Posts: 276
23 Oct 2018, 23:13
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\(a_n\) = \(\frac{n+1}{n+3}\) - \(\frac{n}{n+2}\)
\(a_1\) = \(\frac{2}{4}\) - \(\frac{1}{3}\)
\(a_2\) = \(\frac{3}{5}\) - \(\frac{2}{4}\)
\(a_3\) = \(\frac{4}{6}\) - \(\frac{3}{5}\)
\(a_4\) = \(\frac{5}{7}\) - \(\frac{4}{6}\)
.
.
.
.
.
\(a_{58}\) = \(\frac{59}{61}\) - \(\frac{58}{60}\)
\(a_{59}\) = \(\frac{60}{62}\) - \(\frac{59}{61}\)
\(a_{60}\) = \(\frac{61}{63}\) - \(\frac{60}{62}\)
All the terms get canceled out and only \(\frac{61}{63}\) - \(\frac{1}{3}\) remains
\(\frac{61}{63}\) - \(\frac{1}{3}\) = \(\frac{40}{63}\)
OPTION : D
\(a_1\) = \(\frac{2}{4}\) - \(\frac{1}{3}\)
\(a_2\) = \(\frac{3}{5}\) - \(\frac{2}{4}\)
\(a_3\) = \(\frac{4}{6}\) - \(\frac{3}{5}\)
\(a_4\) = \(\frac{5}{7}\) - \(\frac{4}{6}\)
.
.
.
.
.
\(a_{58}\) = \(\frac{59}{61}\) - \(\frac{58}{60}\)
\(a_{59}\) = \(\frac{60}{62}\) - \(\frac{59}{61}\)
\(a_{60}\) = \(\frac{61}{63}\) - \(\frac{60}{62}\)
All the terms get canceled out and only \(\frac{61}{63}\) - \(\frac{1}{3}\) remains
\(\frac{61}{63}\) - \(\frac{1}{3}\) = \(\frac{40}{63}\)
OPTION : D
General Discussion
24 Oct 2018, 03:49
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Solution
Given:
- • In a series, \(a_n = \frac{(n + 1)}{(n + 3)} – \frac{n}{(n + 2)}\)
To find:
- • The sum of the first 60 terms of this series
Approach and Working:
- • \(a_n = \frac{(n + 1)}{(n + 3)} – \frac{n}{(n + 2)} = \frac{[(n + 1)(n + 2) – n(n + 3)]}{(n + 3)(n + 2)} = \frac{2}{(n + 3)(n + 2)} = 2[\frac{1}{(n + 2)} – \frac{1}{(n + 3)}]\)
• Thus,
- o \(a_1 = 2(\frac{1}{3} – \frac{1}{4})\)
o \(a_2 = 2(\frac{1}{4} – \frac{1}{5})\)
o \(a_3 = 2(\frac{1}{5} – \frac{1}{6})\)
o .
o .
o .
o \(a_{59} = 2(\frac{1}{61} – \frac{1}{62})\)
o \(a_{60} = 2(\frac{1}{62} – \frac{1}{63})\)
Hence, the correct answer is Option D
Answer: D
