Let α , β be the roots of the quadratic equation x 2 + 6 x + 3 = 0 . Then α 23 + β 23 + α 14 + β 14 α 15 + β 15 + α 10 + β 10 is equal to

Step-by-step solution

To find the value of α 23 + β 23 + α 14 + β 14 α 15 + β 15 + α 10 + β 10 , Let a n = α n + β n Hence, α 23 + β 23 + α 14 + β 14 α 15 + β 15 + α 10 + β 10 = a 23 + a 14 a 15 + a 10 Now, x 2 + 6 x + 3 = 0 has roots α   &   β So, x = - 6 ± - 6 2 ⇒ x = 6 - 1 ± i 2 ⇒ x = 3 - 1 ± i 2 Hence, α = 3   e i 3 π 4 and β = 3 e i 5 π 4 Now, solving α 23 + &#946