KVPY SX SB Exam 2012 Question Paper

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Question : 86 of 120

Marks: +1, -0

Define a sequence

(a_n)

by

a_1=5,a_n=a_1a_2....a_{n-1}+4

for

n > 1

. Then

\lim↙{n→∞}√a_n/{a_{n-1}}

equals

$1/2$
$2/5$
1
does not exist

Solution:

a_1 = 5

a_n = a_1a_2 ….. a_{n-1} + 4

a_2 = a_1 + 4 = 9

a_3 = a_1 .a_2 + 4

= 5 ×9 + 4 = 49

a_4 = a_1a_2a_3 + 4 = 2209

a_5 = a_1a_2a_3a_4 + 4

= 4870849= (2207)^2

a_5 = (a_4- 2)^2

a_4 = (49 - 2)^2 = (a_3 -2)^2

a_3 = (9-2)^2 = (a^2 -2)^2

a_n = (a_{n-1} -2)^2

√a_n=a_{n-1}-2

√a_n/a_{n-1}={a_{n-1}-2}/a_{n-1}

=1-\lim↙{n→∞}2/a_{n-1}

∵\lim↙{n→∞}a_{n-1}=∞=1

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