Test Index

CAT Exam Model Paper 4 with solutions for free online practice

Question : 9 of 60

Marks: +1, -0

If S

= \frac{1}{100} + \frac{1}{101} + \frac{1}{102} + \dots + \frac{1}{1000}

then

Solution:

S = \frac{1}{100} + \frac{1}{101} + \frac{1}{102} + \dots + \frac{1}{998} + \frac{1}{999} + \frac{1}{1000}

= \left( \frac{1}{100} + \frac{1}{1000} \right) + \left( \frac{1}{101} + \frac{1}{999} \right) + \dots(450\text{ terms}) + \frac{1}{550}

= \frac{1000+100}{100 \times 1000} + \frac{999+101}{101 \times 999} + \dots + \frac{1}{550}

= \frac{1100}{100 \times 1000} + \frac{1100}{101 \times 999} + \dots + \frac{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 > \frac{1100}{550 \times 550} + \frac{1100}{550 \times 550} + \dots(450\text{ terms}) + \frac{1}{550}

S > 450 \left( \frac{1100}{550 \times 550} \right) + \frac{1}{550} > 1

Go to Question:

Prev Question

Next Question