If $1,\;{\log _y}x,\;{\log _z}y,\; - 15{\log _x}z$ are in $A.P.$, then

A series whose $n^{th}$ term is $\left( {\frac{n}{x}} \right) + y,$ the sum of $r$ terms will be

If $\tan \left(\frac{\pi}{9}\right), x, \tan \left(\frac{7 \pi}{18}\right)$ are in arithmetic progression and $\tan \left(\frac{\pi}{9}\right), y, \tan \left(\frac{5 \pi}{18}\right)$ are also in arithmetic progression, then $|x-2 y|$ is equal to:

Insert five numbers between $8$ and $26$ such that resulting sequence is an $A.P.$

Suppose $a_{1}, a_{2}, \ldots, a_{ n }, \ldots$ be an arithmetic progression of natural numbers. If the ratio of the sum of the first five terms of the sum of first nine terms of the progression is $5: 17$ and $110< a_{15} < 120$ , then the sum of the first ten terms of the progression is equal to -

If $a,\;b,\;c$ are in $A.P.$, then $\frac{{{{(a - c)}^2}}}{{({b^2} - ac)}} = $