If $a,\;b,\;c,\;d$ and $p$ are different real numbers such that $({a^2} + {b^2} + {c^2}){p^2} - 2(ab + bc + cd)p + ({b^2} + {c^2} + {d^2}) \le 0$, then $a,\;b,\;c,\;d$ are in

Find the sum to indicated number of terms in each of the geometric progressions in $\left.1,-a, a^{2},-a^{3}, \ldots n \text { terms (if } a \neq-1\right)$

The numbers $(\sqrt 2 + 1),\;1,\;(\sqrt 2 - 1)$ will be in

For $0<\mathrm{c}<\mathrm{b}<\mathrm{a}$, let $(\mathrm{a}+\mathrm{b}-2 \mathrm{c}) \mathrm{x}^2+(\mathrm{b}+\mathrm{c}-2 \mathrm{a}) \mathrm{x}$ $+(c+a-2 b)=0$ and $\alpha \neq 1$ be one of its root. Then, among the two statements

$(I)$ If $\alpha \in(-1,0)$, then $\mathrm{b}$ cannot be the geometric mean of $\mathrm{a}$ and $\mathrm{c}$

$(II)$ If $\alpha \in(0,1)$, then $\mathrm{b}$ may be the geometric mean of $a$ and $c$

If $x,\,2x + 2,\,3x + 3,$are in $G.P.$, then the fourth term is

If $n$ geometric means between $a$ and $b$ be ${G_1},\;{G_2},\;.....$${G_n}$ and a geometric mean be $G$, then the true relation is