Sequence of series practice · Mixed difficulty

$A_r, r=1,2,3, \ldots, n$ are $n$ points on the parabola $y^2=4 x$ in the first quadrant. If $A_r=\left(x_r, y_r\right)$, where $x_1, x_2, x_3, \ldots, x_n$ are in GP and $x_1=1$, $x_2=2$, then $y_n$ is equal to

The set of real values of $x$ for which $\log _{0.2} \frac{x+2}{x} \leq 1$ is

Let $\alpha ,\beta ,\gamma \;\mathrm{and}\;\delta$ are four positive real numbers such that their product is unity, then the least value of $\left( {1 + \mathrm{\alpha }} \right)\left( {1 + \mathrm{\beta }} \right)\left( {1 + \mathrm{\gamma }} \right)\left( {1 + \mathrm{\delta }} \right)$ is

Let ${S_1},\;{S_2}, \ldots$ be squares such that for each $n \ge 1,$ the length of a side of ${S_n}$ equals the length of a diagonal of ${S_{n + 1}}.$ If the length of a side of ${S_1}$ is 10cm, then for which of the following values of $n$ is the area of ${S_n}$ less than 1 sq cm?

Which of the following is harmonic mean of numbers $\frac{1}{3}$ and $\frac{1}{7}$ ?

If $(1+3+5+\ldots+p)+(1+$ $3+5+\ldots+q)=(1+3+5+\ldots+r)$ where each set of parentheses contains the sum of consecutive odd integers as shown, the smallest possible value of $p+q+r$, (where $p>6$) is

If $S_n=n P+\frac{n(n-1)}{2} Q$, where $S_n$ denotes the sum of the first $n$ terms of an A.P., then the common difference is

The sum of the series $1+2 x+3 x^2+4 x^3+\ldots$ (where, $x$ lies between 0 and 1 , i.e. $0 < x < 1$ ) is