Question-43971

Question Number 43971 by LYCON TRIX last updated on 19/Sep/18

Answered by MJS last updated on 19/Sep/18

2 x^{2} + x = 3 y^{2} + y

x = y + a

y^{2} - 4 a y - a (2 a + 1) = 0

y = 2 a \pm \sqrt{a (6 a + 1)}

\sqrt{a (6 a + 1)} integer \Rightarrow a (6 a + 1) = p^{2} \Rightarrow a = q^{2} \Rightarrow

\Rightarrow \sqrt{6 q^{2} + 1} integer

q_{1} = 0 \Rightarrow x_{1} = y_{1} = 0

q_{2} = 2 \Rightarrow x_{2} = 22 \land y_{2} = 18

q_{3} = 20 \Rightarrow x_{3} = 2180 \land y_{3} = 1780

q_{4} = 198 \Rightarrow x_{4} = 213642 \land y_{4} = 174438

\underset{i = 1}{\sum^{4}} (x_{i} + y_{i}) = 392080

Commented by MJS last updated on 19/Sep/18

q_{5} = 1960

Commented by MJS last updated on 19/Sep/18

q_{1} = 0

q_{2} = 2

n ⩾ 2 : q_{n + 1} = 10 q_{n} - q_{n - 1}

Commented by MJS last updated on 19/Sep/18

I think not all questions there hold prizes .

some are too easy

Commented by LYCON TRIX last updated on 20/Sep/18

Yes, we claim cash prize on selected

questions only, {it}^{'} s mentioned in community