Question

If a, ß are the roots of
x² + px +q = 0 and
An = alpha.n+Beta . n, then​

Answer 1

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Answer 2

Since, α and β are the roots of x 2 +px+q=0

⇒α+β=−p....(1)

Since, α and β are the roots of x 2n +p n x n +q n =0

α 2n +p n α n +q n =0 ...(2)

β 2n +p nβ n +q n =0 ....(3)

Subtracting (3) from (2), we get

α 2n −β 2n +p n (α n −β n )=0

⇒α n +β n =−p n ....(4)

Now, since, ( βα ) is the root of x n +1+(x+1) n =0

( β/α ) n +1+( β/α +1) n =0

⇒α n +β n +(α+β) n =0 ...(6)

So, (5) and (6) can be written as −p n +(−p) n =0 .... (by (1) and (4))which is possible only when n is an even integer.