• # question_answer If ${{S}_{1}},\ {{S}_{2}},\ {{S}_{3}},...........{{S}_{m}}$ are the sums of $n$ terms of $m$ A.P.'s whose first terms are $1,\ 2,\ 3,\ ...............,m$ and common differences are $1,\ 3,\ 5,\ ...........2m-1$ respectively, then ${{S}_{1}}+{{S}_{2}}+{{S}_{3}}+.......{{S}_{m}}=$ A) $\frac{1}{2}mn(mn+1)$ B) $mn(m+1)$ C) $\frac{1}{4}mn(mn-1)$ D) None of the above

Here $a=1,\ 2,\ 3,\,........,m;\ \ \ d=1,\ 3,\ 5,........,2m-1$ and $n=n$, then ${{S}_{1}}+{{S}_{2}}+.......+{{S}_{m}}=\frac{1}{2}mn(mn+1)$ $\left[ \text{Using}\ S\ =\frac{m}{2}(a+l).\ \text{Since}\ {{S}_{1}},\ {{S}_{2}},\ {{S}_{3}},......{{S}_{m}}\ \text{form}\ \text{an}\ \text{A}\text{.P}\text{.} \right]$