Determine $k$ so that $\frac{2}{3}, k$ and $\frac{5}{8} k$ are the three consecutive terms of an $A.P.$

The first term of an $A.P.$ is $2$ and the common difference is $4$. The sum of its $40$ terms will be:

If the sum of the first $11$ terms of an $A.P.$,$a_{1}, a_{2}, a_{3}, \ldots$ is $0$ $(a_{1} \neq 0)$,then the sum of the $A.P.$,$a_{1}, a_{3}, a_{5}, \ldots, a_{23}$ is $k a_{1}$,where $k$ is equal to:

Terms $a, 1, b$ are in arithmetic progression and terms $1, a, b$ are in geometric progression. Find $a$ and $b$ (given $a \neq b$).

The geometric series $a + ar + ar^2 + ar^3 + \dots \infty$ has a sum of $7$,and the sum of the terms involving odd powers of $r$ is $3$. Then,the value of $(a^2 - r^2)$ is:

The interior angles of a polygon are in $A.P.$ If the smallest angle is $120^{\circ}$ and the common difference is $5^{\circ}$,then the number of sides is