Let ${T_r}$ be the ${r^{th}}$ term of an $A.P.$ for $r = 1, 2, 3, ....$. If for some positive integers $m, n$ we have ${T_m} = \frac{1}{n}$ and ${T_n} = \frac{1}{m}$,then ${T_{mn}}$ equals

$0.5737373...... = $

For a series $S = 1 - 2 + 3 - 4 + \dots$ up to $n$ terms,
Statement $-1$: The sum of the series is always dependent on the value of $n$,i.e.,whether it is even or odd.
Statement $-2$: The sum of the series is $-\frac{n}{2}$ when $n$ is any even integer.

If $\frac{S_n}{S_m} = \frac{n^4}{m^4}$ (where $S_k$ is the sum of the first $k$ terms of an $A$.$P$. $a_1, a_2, \dots, \infty$),then the value of $\frac{a_{m+1}}{a_{n+1}}$ in terms of $m$ and $n$ will be

The sum of the first $20$ terms of the series $1 + \frac{3}{2} + \frac{7}{4} + \frac{15}{8} + \frac{31}{16} + \dots$ is?

Statement $-1$: The sum of the series $1 + (1 + 2 + 4) + (4 + 6 + 9) + (9 + 12 + 16) + \dots + (361 + 380 + 400)$ is $8000$.
Statement $-2$: $\sum_{k=1}^{n} (k^3 - (k-1)^3) = n^3$,for any natural number $n$.