The geometric and harmonic means of two numbers $x_1$ and $x_2$ are $18$ and $16\frac{8}{13}$ respectively. The value of $|x_1 - x_2|$ is

If the $p^{th}$,$q^{th}$,and $r^{th}$ terms of an arithmetic sequence are $a$,$b$,and $c$ respectively,then the value of $[a(q - r) + b(r - p) + c(p - q)]$ is:

For any three positive real numbers $a, b, c$,if $9(25a^2 + b^2) + 25(c^2 - 3ac) = 15b(3a + c)$,then:

If $p$ times the $p^{th}$ term of an $A.P.$ is equal to $q$ times the $q^{th}$ term of an $A.P.$,then the $(p + q)^{th}$ term is

Given $x = 1 + a + a^2 + ... \infty$ $(a < 1)$ and $y = 1 + b + b^2 + ... \infty$ $(b < 1)$,find the value of $1 + ab + a^2b^2 + ... \infty$.

Let $a_{n}$ be the $n^{\text{th}}$ term of a $G$.$P$. of positive terms. If $\sum_{n=1}^{100} a_{2n+1} = 200$ and $\sum_{n=1}^{100} a_{2n} = 100$,then $\sum_{n=1}^{200} a_{n}$ is equal to: