CAT Exam Model Paper 4 with solutions for free online practice

Question : 9 of 60

Marks: +1, -0

If S

= { 1 } /{ 100 } + { 1 }/ { 101 } + { 1 }/ { 102 } + ... + { 1 }/ { 1000 }

then

Solution:

{ S } = { 1 }/ { 100 } + { 1 }/ { 101 } + { 1 }/{ 102 } +... + \ { 1 } /{ 998 } + { 1 }/ { 999 } + { 1 } /{ 1000 }

= ( { 1 }/ { 100 } + { 1 } /{ 1000 } ) + ( { 1 }/ { 101 } + { 1 }/ { 999 } ) + ...(450 \terms)+ { 1 }/ { 550 }

= { 1000 + 100 }/ { 100 × 1000 } + { 999 + 101 } /{ 101 × 999 } +....+ { 1 }/ { 550 }

= { 1100 } /{ 100 × 1000 } + { 1100 } /{ 101 ×999 } + ... + { 1 }/ { 550 }

In the first 450 terms, the sum of the two numbers in the denominator is constant. Thus, their product will be maximum when they are equal
∴

{ S } > { 1100 } /{ 550 × 550 } + { 1100 }/ { 550 × 550 } + ...(450 \terms)+ { 1 } /{ 550 }

{ S } > 450 ( { 1100 } /{ 550 × 550 } ) + { 1 }/ { 550 } > 1

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