If $n$ is the smallest natural number such that $n+2 n+3 n+\ldots+99 n$ is a perfect square, then the number of digits of $n^2$ is

The sum of the first four terms of an $A.P.$ is $56$. The sum of the last four terms is $112$. If its first term is $11$, the number of terms is

If the first term of an $A.P.$ is $3$ and the sum of its first $25$ terms is equal to the sum of its next $15$ terms, then the common difference of this $A.P.$ is :

If $\frac{{3 + 5 + 7 + ..........{\rm{to}}\;n\;{\rm{terms}}}}{{5 + 8 + 11 + .........{\rm{to}}\;10\;{\rm{terms}}}} = 7$, then the value of $n$ is

What is the sum of all two digit numbers which give a remainder of $4$ when divided by $6$ ?

Suppose the sum of the first $m$ terms of an arithmetic progression is $n$ and the sum of its first $n$ terms is $m$, where $m \neq n$. Then, the sum of the first $(m+n)$ terms of the arithmetic progression is