Try this beautiful Objective Limit Problem appeared in ISI Entrance - 2021.
Let us denote the fractional part of
a real number \(x\) by \(\{x\}.\)
(Note \({x}=x-[x]\) where \([x]\)
is the integer part of \(x\).)
Then
\[
\lim _{n \rightarrow \infty}{(3+2 \sqrt{2})^{n}}.
\]
(A) equals 0
(B) equals 1
(C) equals \(\frac{1}{2}\)
(D) does not exist
Fractional part of a real number
Greatest Integer function
IIT mathematics by Asit Das Gupta
ISI UG Entrance - 2021 , Objective problem number - 23
(B) equals 1
Try to find the fractional part of \((3+2 \sqrt{2})^{n} = N \)(Let) .
\(\bullet \) Observe \((3+2 \sqrt{2})^{n} + (3-2 \sqrt{2})^{n}\) is an integer.
\(\bullet \) And use \( 0 < (3-2 \sqrt{2}) < 1.\)
\(\bullet \) Obviuosly \(0<(3-2 \sqrt{2})^{n}<1.\)
\(\bullet \) Also assume , \(p=(3-2 \sqrt{2})^{n}.\)
\(\bullet \) As \(N+p = integer = [N] + \{N\} + p ,\)
so \( \{N\} + p = integer - [N]= integer.\)
\(\bullet \) Hence proceed.
\(\bullet \) It is very easy to find that \( \{N\} + p =1.\)
$\lim _{n \rightarrow \infty}(3+2 \sqrt{2})^{n}$
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