"Let $\alpha$ and $\beta$ be two real numbers such that $\alpha$ + $\beta$ = 1 and $\alpha$$\beta$ = $-$1. Let pn = ($\alpha$)n + ($\beta$)n, pn$-$1 = 11 and pn+1 = 29 for some integer n $\ge$ 1. Then, the value of p$_n^2$ is ___________."

Question

Accepted Answer

"Given, $\alpha$ + $\beta$ = 1, $\alpha$$\beta$ = $-$ 1

$\therefore$ Quadratic equation with roots $\alpha$, $\beta$ is x² $-$ x $-$ 1 = 0

$\Rightarrow$ $\alpha$² = $\alpha$ + 1

Multiplying both sides by $\alpha$^n$-$1

$\alpha$^n$+$1 = $\alpha$ⁿ + $\alpha$^n$-$1 ......(1)

Similarly,

$\beta$^{n + 1} = $\beta$ⁿ + $\beta$^{n + 1} ..... (2)

Adding (1) & (2)

$${\alpha ^{n + 1}} + {\beta ^{n + 1}} = ({\alpha ^n} + {\beta ^n}) + ({\alpha ^{n - 1}} + {\beta ^{n - 1}})$$

$\Rightarrow$ P_n+1 = P_n + P_n$-$1

$\Rightarrow$ 29 = P_n + 11 (Given, P_{n + 1} = 29, P_{n $-$ 1} = 11)

$\Rightarrow$ P_n = 18

$\therefore$ $P_n^2$ = 18² = 324"

Let $\alpha$ and $\beta$ be two real numbers such that $\alpha$ + $\beta$ = 1 and $\alpha$$\beta$ = $-$1. Let p_n = ($\alpha$)ⁿ + ($\beta$)ⁿ, p_n$-$1 = 11 and p_n+1 = 29 for some integer n $\ge$ 1. Then, the value of p$_n^2$ is ___________.

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Let $\alpha$ and $\beta$ be two real numbers such that $\alpha$ + $\beta$ = 1 and $\alpha$$\beta$ = $-$1. Let pn = ($\alpha$)n + ($\beta$)n, pn$-$1 = 11 and pn+1 = 29 for some integer n $\ge$ 1. Then, the value of p$_n^2$ is ___________.

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Let $\alpha$ and $\beta$ be two real numbers such that $\alpha$ + $\beta$ = 1 and $\alpha$$\beta$ = $-$1. Let p_n = ($\alpha$)ⁿ + ($\beta$)ⁿ, p_n$-$1 = 11 and p_n+1 = 29 for some integer n $\ge$ 1. Then, the value of p$_n^2$ is ___________.