The sum of the series $3 + 33 + 333 + ... + n$ terms is
$\frac{1}{{27}}({10^{n + 1}} + 9n - 28)$
$\frac{1}{{27}}({10^{n + 1}} - 9n - 10)$
$\frac{1}{{27}}({10^{n + 1}} + 10n - 9)$
None of these
The sum of $100$ terms of the series $.9 + .09 + .009.........$ will be
For a sequence $ < {a_n} > ,\;{a_1} = 2$ and $\frac{{{a_{n + 1}}}}{{{a_n}}} = \frac{1}{3}$. Then $\sum\limits_{r = 1}^{20} {{a_r}} $ is
Let $\left\{a_k\right\}$ and $\left\{b_k\right\}, k \in N$, be two G.P.s with common ratio $r_1$ and $r_2$ respectively such that $a_1=b_1=4$ and $r_1 < r_2$. Let $c_k=a_k+k, \in N$. If $c_2=5$ and $c_3=13 / 4$ then $\sum \limits_{k=1}^{\infty} c_k - \left(12 a _6+8 b _4\right)$ is equal to
Suppose the sides of a triangle form a geometric progression with common ratio $r$. Then, $r$ lies in the interval
If $a,\;b,\;c,\;d$ and $p$ are different real numbers such that $({a^2} + {b^2} + {c^2}){p^2} - 2(ab + bc + cd)p + ({b^2} + {c^2} + {d^2}) \le 0$, then $a,\;b,\;c,\;d$ are in