If ${(1 - x + {x^2})^n} = {a_0} + {a_1}x + {a_2}{x^2} + .... + {a_{2n}}{x^{2n}}$, then ${a_0} + {a_2} + {a_4} + .... + {a_{2n}} = $
$\frac{{{3^n} + 1}}{2}$
$\frac{{{3^n} - 1}}{2}$
$\frac{{1 - {3^n}}}{2}$
${3^n} + \frac{1}{2}$
The sum of the coefficients in the expansion of ${(1 + x - 3{x^2})^{3148}}$ is
The sum of the series $aC_0 + (a + b)C_1 + (a + 2b)C_2 + ..... + (a + nb)C_n$ is where $Cr's$ denotes combinatorial coefficient in the expansion of $(1 + x)^n, n \in N$
The sum to $(n + 1)$ terms of the following series $\frac{{{C_0}}}{2} - \frac{{{C_1}}}{3} + \frac{{{C_2}}}{4} - \frac{{{C_3}}}{5} + $..... is
The sum of the series $\sum\limits_{r = 0}^n {{{( - 1)}^r}\,{\,^n}{C_r}\left( {\frac{1}{{{2^r}}} + \frac{{{3^r}}}{{{2^{2r}}}} + \frac{{{7^r}}}{{{2^{3r}}}} + \frac{{{{15}^r}}}{{{2^{4r}}}} + .....m\,{\rm{terms}}} \right)} $ is
The number $111......1 $ ( $ 91$ times) is