KVPY SX SB Exam 05-Nov-2017 Question Paper

Question : 20 of 120

Marks: +1, -0

Let

a_i = i + 1/i \for i = 1,2, ......., 20.

Put

p =1/20 (a_1 + a_2 + ....... + a_20)

and

q = 1/20 (1/a_1 + 1/a_2 + ....... + 1/a_20)

. Then

Solution:

q > 0, try to the contra prove that

q <{22-p}/21

q+p/21<22/21

q+p/21=1/20[∑↖{20}↙{i=1}1/a_i+1/21∑↖{20}↙{i=1}a_i]

1/20[∑↖{20}↙{i=1}(i/{i^2+2}) +1/21∑↖{20}↙{i=1}(i+1/i)]

1/2+1/20[∑↖{20}↙{i=1}i/{i^2+2} + ∑↖{20}↙{i=1}(i /{21 i})]

1/2+1/20[1/2+∑↖{20}↙{i=1}i/{i^2+2} + ∑↖{20}↙{i=1} i /{21 i} ]

1/2+1/20[1/2+2/5∑↖{20}↙{i=1}1 + 1/21∑↖{20}↙{i=1} 1 ]

1/2+1/20[1/2+2/5×19+1/21×20]

1/2+1/20 (1 + 8 + 1)

<1/2+1/2

<22/21

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