If $^n{C_4},{\,^n}{C_5},$ and ${\,^n}{C_6},$ are in $A.P.,$ then $n$ can be 

If the $10^{\text {th }}$ term of an A.P. is $\frac{1}{20}$ and its $20^{\text {th }}$ term is $\frac{1}{10},$ then the sum of its first $200$ terms is

The sum of the common terms of the following three arithmetic progressions.

$3,7,11,15,...................,399$

$2,5,8,11,............,359$ and

$2,7,12,17,...........,197$, is equal to $................$.

Let $X$ be the set consisting of the first $2018$ terms of the arithmetic progression $1,6,11$,

. . . .and $Y$ be set consisting of the first $2018$ terms of the arithmetic progression $9, 16, 23$,. . . . . Then, the number of elements in the set $X \cup Y$ is. . . . 

In $\Delta ABC$, if $a, b, c$ are in $A.P.$ (with usual notations), identify the incorrect statements -

The sums of $n$ terms of two arithmatic series are in the ratio $2n + 3:6n + 5$, then the ratio of their ${13^{th}}$ terms is