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

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 $P_{96}$, $P_{97}$, $P_{98}$, $P_{99}$.

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

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