Question

2(Xn)²-2Xn = Xn+1 - 1 X1 = 1/2 lim x-->inf (Xn)^a^n = L L is non zero and finite, find a and L

Answer 1

a=0, L=1

Answer 2

The recurrence relation $X_{n+1} = 2X_n^2 - 2X_n + 1$ with $X_1 = 1/2$ yields $X_n = 1/2$ for all $n$. The limit condition $\lim_{n \to \infty} (X_n)^{a^n} = L$ becomes $\lim_{n \to \infty} (1/2)^{a^n} = L$, with $L \neq 0$ and finite. If $a=0$, then $a^n = 0$ for $n \ge 1$, so $(1/2)^0 = 1$, giving $L=1$. If $a=1$, then $a^n = 1$, so $(1/2)^1 = 1/2$, giving $L=1/2$. If $|a|<1$ and $a \neq 0$, then $a^n \to 0$, so $(1/2)^0 = 1$, giving $L=1$. For uniqueness, the case $a=0$ is conventionally chosen, leading to $a=0$ and $L=1$.