The first term of the ${11^{th}}$ group in the following sequence of groups $(1), (2, 3, 4), (5, 6, 7, 8, 9), \dots$ is

If $a, x$ are real numbers and $|a| < 1, |x| < 1$, then $1 + (1+a)x + (1+a+a^2)x^2 + \dots \infty$ is equal to

If $\alpha, \beta$ are natural numbers such that $100^{\alpha} - 199\beta = (100)(100) + (99)(101) + (98)(102) + \ldots + (1)(199)$,then the slope of the line passing through $(\alpha, \beta)$ and the origin is:

The odd numbers are divided as follows:
Row $1$: $1, 3$
Row $2$: $5, 7, 9, 11$
Row $3$: $13, 15, 17, 19, 21, 23$
Then the sum of the $n^{th}$ row is:

If $S_n = 2 + 4 + 7 + 11 + \dots + n$ terms,then $t_n = \dots$

The value of $\sum\limits_{r = 16}^{30} {(r + 2)(r - 3)}$ is equal to