For a given $A.P.$,$S_{10} = 50$ and $a = 0.5$. Then,$d = \ldots \ldots \ldots . .$

The famous mathematician associated with finding the sum of the first $100$ natural numbers is

In an $AP$,if $S_{n} = 3n^{2} + 5n$ and $a_{k} = 164$,find the value of $k$.

$A$ plastic toy consists of $14$ circular discs of different radii. The radius of the smallest disc is $1\, cm$, the radius of the next bigger disc is $2\, cm$, and so on, such that the radius of the biggest disc is $14\, cm$. Find the sum of the circumferences of all the discs in $cm$.

The sum of the first $n$ terms of an $A.P.$ is given by $S_{n} = \ldots \ldots \ldots \ldots$

The sums of $n$ terms,$2n$ terms,and $3n$ terms of an $A.P.$ are $S_1, S_2,$ and $S_3$ respectively. Prove that $S_3 = 3(S_2 - S_1)$.