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# ISI 2021 Objective Problem 23 I A Problem from Limit

Try this beautiful Objective Limit Problem appeared in ISI Entrance - 2021.

## Problem

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

### Key Concepts

Fractional part of a real number

Greatest Integer function

## Suggested Book | Source | Answer

IIT mathematics by Asit Das Gupta

ISI UG Entrance - 2021 , Objective problem number - 23

(B) equals 1

## Try with Hints

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 $$\{N\} + p =1.$$

∙ Therefore ,

$\lim _{n \rightarrow \infty}(3+2 \sqrt{2})^{n}$

=limn→∞{N}=limn→∞(1p)=1

[As,limn→∞p=limn→∞(3−22)n=0]

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