For each positive integer n , let A_n=\max{\{(n /r)∣ 0 \leq r \leqn\} . Then the number of elements n in {1 ,2 ,...,20 } for which 1.9 \leq A_n /A_{n-1}

KVPY SX SB Exam 20141 Question Paper

Question : 3 of 120

Marks: +1, -0

For each positive integer n , let

A_n=\max{\{(n /r)∣ 0 ≤ r ≤n\}

. Then the number of elements n in {1 ,2 ,...,20 } for which

1.9 ≤ A_n /A_{n-1} < 2

Solution:

Case (1) n = even

A_n/{A _{n-1}}={^nC_{n\/2}}/{^{n-1}C_{{n-1-1}/2}}=2

so for all n even given relation is true.
Case (2) n = odd

A_n/{A _{n-1}}={^nC_{{n-1}/2}}/{^{n-1}C_{{n-1 }/2}}={2n}/{n+1}

which satisfies only for n = 19

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