If $a, b, c, d, e$ are in $A.P.$,then the value of $a + b + 4c - 4d + e$ in terms of $a$,if possible,is:

The number of terms common to the two $A$.$P$.'s $3, 7, 11, \ldots, 407$ and $2, 9, 16, \ldots, 709$ is

If $1, \log_{y}x, \log_{z}y, -15\log_{x}z$ are in $A.P.$,then

Let $a_1, a_2, \ldots, a_n$ be in an $A$.$P$. If $a_5 = 2a_3$ and $a_{11} = 18$,then $12\left(\frac{1}{\sqrt{a_{10}}+\sqrt{a_{11}}} + \frac{1}{\sqrt{a_{11}}+\sqrt{a_{12}}} + \ldots + \frac{1}{\sqrt{a_{17}}+\sqrt{a_{18}}}\right)$ is equal to $..........$.

The sides of a right-angled triangle are in arithmetic progression. If the triangle has an area of $24$,then what is the length of its smallest side?

If the roots of the equation $32x^3 - 48x^2 + 22x - 3 = 0$ are in arithmetic progression,then the square of the common difference of the roots is