After inserting $n$, $A.M.'s$ between $2$ and $38$, the sum of the resulting progression is $200$. The value of $n$ is

If the sum of the first $2n$ terms of $2,\,5,\,8...$ is equal to the sum of the first $n$ terms of $57,\,59,\,61...$, then $n$ is equal to

If the sum of the series $54 + 51 + 48 + .............$ is $513$, then the number of terms are

If $p,\;q,\;r$ are in $A.P.$ and are positive, the roots of the quadratic equation $p{x^2} + qx + r = 0$ are all real for

If $\log _{10} 2, \log _{10} (2^x + 1), \log _{10} (2^x + 3)$ are in $A.P.,$ then :-

Write the first three terms in each of the following sequences defined by the following:

$a_{n}=\frac{n-3}{4}$