If α , β are roots of the equation x 2 + 5 2 x + 10 = 0 , α β and P n = α n - β n for each positive integer n , then the value of P 17 P 20 + 5 2 P 17 P 19 P 18 P 19 + 5 2 P 18 2 is equal to

Question

If α , β are roots of the equation x 2 + 5 2 x + 10 = 0 , α > β and P n = α n - β n for each positive integer n , then the value of P 17 P 20 + 5 2 P 17 P 19 P 18 P 19 + 5 2 P 18 2 is equal to

TestHub · Accepted Answer

Given, x 2 + 5 2 x + 10 = 0 and P n = α n - β n Now P 17 P 20 + 5 2 P 17 P 19 P 18 P 19 + 5 2 P 18 2 = P 17 P 20 + 5 2 P 19 P 18 P 19 + 5 2 P 18 = P 17 α 20 - β 20 + 5 2 α 19 - β 19 P 18 α 19 - β 19 + 5 2 α 18 - β 18 = P 17 α 19 α + 5 2 - β 19 β + 5 2 P 18 α 18 α + 5 2 - β 18 β + 5 2 Since α + 5 2 = - 10 / α . . . 1 & β + 5 2 = - 10 / β . . . 2 Now put these values in above expression P 17 α 19 α + 5 2 - β 19 β + 5 2 P 18 α 18 α + 5 2 - β 18 β + 5 2 = - 10 P 17 P 18 - 10 P 18 P 17 = 1

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