If $x$ is real, then the value of $\frac{{{x^2} + 34x - 71}}{{{x^2} + 2x - 7}}$ does not lie between

Let $\alpha$ and $\beta$ be the roots of $x^2-x-1=0$, with $\alpha>\beta$. For all positive integers $n$, define

$a_n=\frac{\alpha^n-\beta^n}{\alpha-\beta}, n \geq 1$

$b_1=1 \text { and } b_n=a_{n-1}+a_{n+1}, n \geq 2.$

Then which of the following options is/are correct?

$(1)$ $a_1+a_2+a_3+\ldots . .+a_n=a_{n+2}-1$ for all $n \geq 1$

$(2)$ $\sum_{n=1}^{\infty} \frac{ a _{ n }}{10^{ n }}=\frac{10}{89}$

$(3)$ $\sum_{n=1}^{\infty} \frac{b_n}{10^n}=\frac{8}{89}$

$(4)$ $b=\alpha^n+\beta^n$ for all $n>1$

If $x$ is real, then the maximum and minimum values of the expression $\frac{{{x^2} - 3x + 4}}{{{x^2} + 3x + 4}}$ will be

If $72^x \cdot 48^y=6^{x y}$, where $x$ and $y$ are non-zero rational numbers, then $x+y$ equals

The smallest value of ${x^2} - 3x + 3$ in the interval $( - 3,\,3/2)$ is

Suppose $a, b, c$ are positive integers such that $2^a+4^b+8^c=328$. Then, $\frac{a+2 b+3 c}{a b c}$ is equal to