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# A series is defined, for positive integer n, by an=n+1n+3−nn+2 What

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Math Expert
Joined: 02 Sep 2009
Posts: 58458
A series is defined, for positive integer n, by an=n+1n+3−nn+2 What  [#permalink]

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23 Oct 2018, 22:03
00:00

Difficulty:

55% (hard)

Question Stats:

67% (02:41) correct 33% (03:41) wrong based on 56 sessions

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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

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Re: A series is defined, for positive integer n, by an=n+1n+3−nn+2 What  [#permalink]

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23 Oct 2018, 23:13
$$a_n$$ = $$\frac{n+1}{n+3}$$ - $$\frac{n}{n+2}$$

$$a_1$$ = $$\frac{2}{4}$$ - $$\frac{1}{3}$$

$$a_2$$ = $$\frac{3}{5}$$ - $$\frac{2}{4}$$

$$a_3$$ = $$\frac{4}{6}$$ - $$\frac{3}{5}$$

$$a_4$$ = $$\frac{5}{7}$$ - $$\frac{4}{6}$$
.
.
.
.
.
$$a_{58}$$ = $$\frac{59}{61}$$ - $$\frac{58}{60}$$

$$a_{59}$$ = $$\frac{60}{62}$$ - $$\frac{59}{61}$$

$$a_{60}$$ = $$\frac{61}{63}$$ - $$\frac{60}{62}$$

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
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Re: A series is defined, for positive integer n, by an=n+1n+3−nn+2 What  [#permalink]

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24 Oct 2018, 03:49

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 $$a_{60} = 2(\frac{1}{62} – \frac{1}{63})$$
• Therefore, $$S = 2(\frac{1}{3} – \frac{1}{4} + \frac{1}{4} – \frac{1}{5} + \frac{1}{5} – \frac{1}{6} + …... + \frac{1}{61} – \frac{1}{62} + \frac{1}{62} – \frac{1}{63}) = 2(\frac{1}{3} – \frac{1}{63}) = 2 * \frac{20}{63} = \frac{40}{63}$$

Hence, the correct answer is Option D

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Re: A series is defined, for positive integer n, by an=n+1n+3−nn+2 What   [#permalink] 24 Oct 2018, 03:49
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