Let α and β be the roots of the equation x2-x-1=0. If pk=(α)k+(β)k, k ge 1, then which one of the following statements is not true ?

Question

Let α and β be the roots of the equation x2-x-1=0. If pk=(α)k+(β)k, k ge 1, then which one of the following statements is not true ? (A) (p1+p2+p3+p4+p5)=26 (B) p5=11 (C) p5=p2⋅ p3 (D) p3=p5-p4

Tardigrade Admin · Accepted Answer

α+β=1, αβ=-1 Pk=αk+βk α2-α-1=0 ⇒ αk-αk-1-αk-2=0 βk-βk-1-βk-2=0 ⇒ Pk=Pk-1+Pk-2 P1-α+β=1 P2=(α+β)2-2αβ=1+2=3 P3=4 P4=7 P5=11

Q. Let $α$ and $β$ be the roots of the equation $x^{2} - x - 1 = 0.$ If $p_{k} = (α)^{k} + (β)^{k}, k \geq 1,$ then which one of the following statements is not true ?

$(p_{1} + p_{2} + p_{3} + p_{4} + p_{5}) = 26$

$p_{5} = 11$

$p_{5} = p_{2} \cdot p_{3}$

$p_{3} = p_{5} - p_{4}$

Q. Let α and β be the roots of the equation x2−x−1=0. If pk​=(α)k+(β)k,k≥1, then which one of the following statements is not true ?

(p1​+p2​+p3​+p4​+p5​)=26

p5​=11

p5​=p2​⋅p3​

p3​=p5​−p4​

α+β=1,αβ=−1 Pk​=αk+βk<br/>α2−α−1=0 ⇒αk−αk−1−αk−2=0 &βk−βk−1−βk−2=0 ⇒Pk​=Pk−1​+Pk−2​<br/>P1​−α+β=1 P2​=(α+β)2−2αβ=1+2=3 P3​=4 P4​=7 P5​=11

Q. Let $α$ and $β$ be the roots of the equation $x^{2} - x - 1 = 0.$ If $p_{k} = (α)^{k} + (β)^{k}, k \geq 1,$ then which one of the following statements is not true ?

$(p_{1} + p_{2} + p_{3} + p_{4} + p_{5}) = 26$

$p_{5} = 11$

$p_{5} = p_{2} \cdot p_{3}$

$p_{3} = p_{5} - p_{4}$