Let - and - be two real numbers such that -1 and -1- Let Pn-n-n- Pn-1-11 and Pn-1-29 for some integer n-1- Then- the value of Pn2 is

Given, α+β=1,ab= 1 ∴ Quadratic equation with roots α,β is x^2 x 1=0 ⇒α^2=α+1 Multiflying both sides by α^n 1 α^n+1=α^n+α^n 1...(1) Similarly, β^n+1=b^n+b^n 1... ...

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Quantitative Aptitude

A.M Greater than or Equal to G.M

Let α and β b...

Question

Let $α$ and $β$ be two real numbers such that $α + β = 1$ and $α β = - 1$ . Let $P_{n} = (α)^{n} + (β)^{n}, P_{n - 1} = 11$ and $P_{n + 1} = 29$ for some integer $n \geq 1.$ Then, the value of $P_{n}^{2}$ is

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Solution

Given, $α + β = 1, a b = - 1$
$∴$ Quadratic equation with roots $α, β$ is $x^{2} - x - 1 = 0$
$\Rightarrow α^{2} = α + 1$
Multiflying both sides by $α^{n - 1}$
$α^{n + 1} = α^{n} + α^{n - 1} . . . (1)$
Similarly,
$β^{n + 1} = b^{n} + b^{n - 1} . . . (2)$
Adding (1) & (2)
$α^{n + 1} + β^{n + 1} = (α^{n} + β^{n}) + (α^{n - 1} + β^{n - 1})$
$\Rightarrow P^{n + 1} = P_{n} + P_{n - 1}$
$\Rightarrow 29 = P_{n} + 11$
$\Rightarrow P_{n} = 18$
$∴ P_{n}^{2} = 18^{2} = 324$

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Let α and β be two real numbers such that α+β=1 and αβ=−1. Let Pn=(α)n+(β)n,Pn−1=11 and Pn+1=29 for some integer n≥1. Then, the value of P2n is

Given, α+β=1,ab=−1 ∴ Quadratic equation with roots α,β is x2−x−1=0 ⇒α2=α+1 Multiflying both sides by αn−1 αn+1=αn+αn−1...(1) Similarly, βn+1=bn+bn−1...(2) Adding (1) & (2) αn+1+βn+1=(αn+βn)+(αn−1+βn−1) ⇒Pn+1=Pn+Pn−1 ⇒29=Pn+11 ⇒Pn=18 ∴P2n=182=324

Let $α$ and $β$ be two real numbers such that $α + β = 1$ and $α β = - 1$ . Let $P_{n} = (α)^{n} + (β)^{n}, P_{n - 1} = 11$ and $P_{n + 1} = 29$ for some integer $n \geq 1.$ Then, the value of $P_{n}^{2}$ is