CAT Exam Model Paper 1 with solutions for free online practice

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Question : 88 of 100

Marks: +1, -0

The sum o f the series is

{ 1 }/ { 1.2 .3 } + { 1 } /{ 3.4 .5 } + { 1 }/ { 5.6 .7 }

e $^{2}$ - 1
log 2 - 1
2 log $_{10}$ 2 - 1
None of the above

Solution:

(d)

S = { 1 } /{ 1.2 .3 } + { 1 }/ { 3.4 .5 } + { 1 } /{ 5.6 .7 } + ...

Here

T _ { n } = { 1 }/ { ( 2 n - 1 ). 2 n . ( 2 n + 1 ) }

= { 1 } /{ 2 } [ { 1 } /{ 2 n - 1 } - { 2 }/ { 2 n } + { 1 }/ { 2 n + 1 } ]

∴

{ T _ { 1 } = { 1 }/ { 2 } [ { 1 } /{ 1 } - { 2 }/ { 2 } + { 1 }/ { 3 } ] }

T _ { 2 } = { 1 }/ { 2 } [ { 1 }/ { 3 } - { 2 }/ { 4 } + { 1 }/ { 5 } ]

{ T _ { 3 } = { 1 }/ { 2 } [ { 1 }/ { 5 } - { 2 }/ { 6 } + { 1 }/ { 7 } ] }

∴ Required sum,

S =

{ 1 }/ { 2 } [ ( { 1 } /{ 1 } - { 2 }/ { 2 } + { 1 }/ { 3 } )+({ 1 }/ { 3 } - { 2 }/ { 4 } + { 1 }/ { 5 })+({ 1 }/ { 5 } - { 2 }/ { 6 } + { 1 }/ { 7 })+...]

⇒2S =

( { 1 } /{ 1 } - { 2 }/ { 2 } + { 1 }/ { 3 } )+({ 1 }/ { 3 } - { 2 }/ { 4 } + { 1 }/ { 5 })+({ 1 }/ { 5 } - { 2 }/ { 6 } + { 1 }/ { 7 })+...

=2/3-2/4+2/5-2/6+2/7-2/8+...

=2[1/3-1/4+1/5-1/6+1/7-1/8..]

⇒ 1+2S = 2

[1-1/2+1/3-1/4+1/5-1/6+1/7-1/8+...]

= 2 \log _ { \e } ( 1 + 1 ) = 2 \log _ { e } 2

[∵Here x = 1 and log

_{(1+x)}

=

x-x^2/2+x^3/3-x^4/4+..

]

S = { 2 \log _ { e } 2 - 1 }/ { 2 } = ( \log _ { e } 2 - { 1 }/ { 2 } )

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