If the roots of ${x^2} + x + a = 0$exceed $a$, then

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 $n>1$

The number of integers $k$ for which the equation $x^3-27 x+k=0$ has at least two distinct integer roots is

Let $a$ ,$b$, $c$ , $d$ , $e$ be five numbers satisfying the system of equations

                            $2a + b + c + d + e = 6$
                            $a + 2b + c + d + e = 12$
                            $a + b + 2c + d + e = 24$
                            $a + b + c + 2d + e = 48$
                            $a + b + c + d + 2e = 96$ ,

then $|c|$ is equal to 

If $3$ distinct real number $a$,$b$,$c$ satisfy $a^2(a + p) = b^2 (b + p) = c^2 (c + p)$ where $p \in R$, then value of $bc + ca + ab$ is

The sum of all the solutions of the equation $(8)^{2 x}-16 \cdot(8)^x+48=0$ is :