Question
If a = 2 + √3, then the value of (a6 +
a4 + a2 + 1)/a3 isSolution
a = 2 + √3 1/a = 1/(2 + √3) = 2 - √3 a + 1/a = 2 + √3 + 2 - √3 a + 1/a = 4(a6 + a4 + a2 + 1)/a3 → (a3 (a3 + a + 1/a + 1/a3))/a3 = a3 + a + 1/a + 1/a3 = (a3 + 1/a3) + (a + 1/a) a + 1/a = 4 On cubing both sides, (a + 1/a)3 = (4)3 a3 + 1/a3 + 3 × (a + 1/a ) = 64 a3 + 1/a3 + 3 × (4) = 64 a3 + 1/a3 + 12 = 64 a3 + 1/a3 = 52 then, a3 + 1/a3 + a+1/a = 52 + 4 = 56
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