If $a$ and $d$ are two complex numbers, then the sum to $(n + 1)$ terms of the following series $a{C_0} - (a + d){C_1} + (a + 2d){C_2} - ........$ is
$\frac{a}{{{2^n}}}$
$na$
$0$
None of these
If ${\left( {1 + x} \right)^n} = {c_0} + {c_1}x + {c_2}{x^2} + {c_3}{x^3} + ...... + {c_n}{x^n}$ , then the value of ${c_0} - 3{c_1} + 5{c_2} - ........ + {( - 1)^n}\,(2n + 1){c_n}$ is
If $(1 + x) (1 + x + x^2) (1 + x + x^2 + x^3) ...... (1 + x + x^2 + x^3 + ...... + x^n)$
$\equiv a_0 + a_1x + a_2x^2 + a_3x^3 + ...... + a_mx^m$ then $\sum\limits_{r\, = \,0}^m {\,\,{a_r}}$ has the value equal to
The value $\sum \limits_{ r =0}^{22}{ }^{22} C _{ r }{ }^{23} C _{ r }$ is $.......$
The sum of the coefficients in the expansion of ${(1 + x - 3{x^2})^{3148}}$ is