Try this beautiful Recurrence Problem based on Chessboard from IOQM 2022 Problem 9, Part - A.
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 $P_{96}$, $P_{97}$, $P_{98}$, $P_{99}$.
Recurrence Relation
Algebra
Shifting Of Origin and Order
Excursion of Mathematics, Challenge and Thrill of Pre-College Mathematics
IOQM 2022, Part-A, Problem 9
The Required Area is 8 units.
$x_{n+1} - y_{n+1}$ = $(-1) (x_{n} - y_{n})$
$\rightarrow$ $x_{n+1} - y_{n+1}$ = $(-1)^{n+1}$ $(x_{1} - y_{1})$
$x_{n+1} + x_{n}=(-1)(x_{n} - y_{n})/2$
$\rightarrow x_{n+1} + x_{n}=(-1)^{n+1}$
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
Try this beautiful Recurrence Problem based on Chessboard from IOQM 2022 Problem 9, Part - A.
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 $P_{96}$, $P_{97}$, $P_{98}$, $P_{99}$.
Recurrence Relation
Algebra
Shifting Of Origin and Order
Excursion of Mathematics, Challenge and Thrill of Pre-College Mathematics
IOQM 2022, Part-A, Problem 9
The Required Area is 8 units.
$x_{n+1} - y_{n+1}$ = $(-1) (x_{n} - y_{n})$
$\rightarrow$ $x_{n+1} - y_{n+1}$ = $(-1)^{n+1}$ $(x_{1} - y_{1})$
$x_{n+1} + x_{n}=(-1)(x_{n} - y_{n})/2$
$\rightarrow x_{n+1} + x_{n}=(-1)^{n+1}$
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
