isi-entrance None Q10

isi-entrance · India · subjective_collection Sequences and series, recurrence and convergence Closed-form expression derivation

Let $\{x_n\}$ be a sequence such that $x_1 = 2$, $x_2 = 1$ and $2x_n - 3x_{n-1} + x_{n-2} = 0$ for $n > 2$. Find an expression for $x_n$.

Let $x_n = ka^n$. Then $2ka^n - 3ka^{n-1} + ka^{n-2} = 0$, giving $2a^2 - 3a + 1 = 0$, so $(2a-1)(a-1) = 0$, thus $a_1 = \frac{1}{2}$, $a_2 = 1$.
$x_n = k_1\left(\frac{1}{2}\right)^n + k_2(1)^n$.
From $x_1 = 2$: $\frac{k_1}{2} + k_2 = 2$. From $x_2 = 1$: $\frac{k_1}{4} + k_2 = 1$.
Solving: $k_1 = 4$, $k_2 = 0$.
$\therefore x_n = 4\left(\frac{1}{2}\right)^n = \frac{1}{2^{n-2}}$.

Let $\{x_n\}$ be a sequence such that $x_1 = 2$, $x_2 = 1$ and $2x_n - 3x_{n-1} + x_{n-2} = 0$ for $n > 2$. Find an expression for $x_n$.

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