Indian Regional Mathematical olympiad problem

Let $\alpha$ and $\beta$ be two roots of the quadratic equation $x^{2}+mx-1 = 0$ ,where $m$ is an odd integer.Let $\lambda_n = \alpha^{n}+\beta^{n}$ for $n\ge 0$ .Prove that for $n \ge 0$ ,

(a) $\lambda_n$ is an integer.

(b) $gcd( \lambda_n ,\lambda_{n+1}) = 1$

#Algebra #CosinesGroup #Goldbach'sConjurersGroup #TorqueGroup #Olympiad

Easy Math Editor

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Comments

We get the relation $\lambda_{n+1}=-m.\lambda_n+\lambda_{n-1}$ .

Since $\lambda_0=2,\lambda_1=-m$ , it follows that $(\lambda_0,\lambda_1)=1$ , because m is an odd integer.Also, since the first two terms are integers and we got a recurrence relation with integer coefficients, it follows that $\lambda_n$ must be an integer.

Firstly, $(\lambda_0,\lambda_1)=1$ .Let $(\lambda_n,\lambda_{n-1})=1$ .Then let's assume $(\lambda_{n+1},\lambda_n)=d,d>1$ .Then, using the relation we would get that $d|\lambda_{n-1}$ , which is clearly a contradiction, so we have proved it by induction.

Bogdan Simeonov - 7 years, 1 month ago

Nice ..I also solved in the same way....

Eddie The Head - 7 years, 1 month ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$