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IOQM 2022 Problem 9 | Part: A | Recurrence and Algebra

Try this beautiful Recurrence Problem based on Chessboard from IOQM 2022 Problem 9, Part - A.

The Problem:


Let $P_{0}=(3,1)$ and define $P_{n+1}=\left(x_{n}, y_{n}\right)$ for $n \geq 0$ by

$x_{n+1}=-\frac{3 x_{n}-y_{n}}{2}$,

$\quad y_{n+1}=-\frac{x_{n}+y_{n}}{2}$

Find the area of the quadrilateral formed by the points $P_{96}$, $P_{97}$, $P_{98}$, $P_{99}$.

Key Concepts


Recurrence Relation

Algebra

Shifting Of Origin and Order

Suggested Book | Source | Answer


Excursion of Mathematics, Challenge and Thrill of Pre-College Mathematics

IOQM 2022, Part-A, Problem 9

The Required Area is 8 units.

Try with Hints


$x_{n+1} - y_{n+1}$ = $(-1) (x_{n} - y_{n})$

$\rightarrow$ $x_{n+1} - y_{n+1}$ = $(-1)^{n+1}$ $(x_{1} - y_{1})$

$x_{n+1} + x_{n}=(-1)(x_{n} - y_{n})/2$

$\rightarrow x_{n+1} + x_{n}=(-1)^{n+1}$

Using Hint 1

Similarly

$y_{n+1} + y_{n}=(-1)^{n+1}$

Using these relations find a pattern and find the value of $P_{96}, P_{97}, P_{98}, P_{99}$

Shift the origin to make the calculations easier

Then write the vertices in the clockwise or anti-clockwise direction and find the required area

Math Olympiad Program at Cheenta

Subscribe to Cheenta at Youtube


Try this beautiful Recurrence Problem based on Chessboard from IOQM 2022 Problem 9, Part - A.

The Problem:


Let $P_{0}=(3,1)$ and define $P_{n+1}=\left(x_{n}, y_{n}\right)$ for $n \geq 0$ by

$x_{n+1}=-\frac{3 x_{n}-y_{n}}{2}$,

$\quad y_{n+1}=-\frac{x_{n}+y_{n}}{2}$

Find the area of the quadrilateral formed by the points $P_{96}$, $P_{97}$, $P_{98}$, $P_{99}$.

Key Concepts


Recurrence Relation

Algebra

Shifting Of Origin and Order

Suggested Book | Source | Answer


Excursion of Mathematics, Challenge and Thrill of Pre-College Mathematics

IOQM 2022, Part-A, Problem 9

The Required Area is 8 units.

Try with Hints


$x_{n+1} - y_{n+1}$ = $(-1) (x_{n} - y_{n})$

$\rightarrow$ $x_{n+1} - y_{n+1}$ = $(-1)^{n+1}$ $(x_{1} - y_{1})$

$x_{n+1} + x_{n}=(-1)(x_{n} - y_{n})/2$

$\rightarrow x_{n+1} + x_{n}=(-1)^{n+1}$

Using Hint 1

Similarly

$y_{n+1} + y_{n}=(-1)^{n+1}$

Using these relations find a pattern and find the value of $P_{96}, P_{97}, P_{98}, P_{99}$

Shift the origin to make the calculations easier

Then write the vertices in the clockwise or anti-clockwise direction and find the required area

Math Olympiad Program at Cheenta

Subscribe to Cheenta at Youtube


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