6Number 6ExplanationGiven a sequence −12,34,−18,316,...-\frac{1}{2}, \frac{3}{4}, -\frac{1}{8}, \frac{3}{16}, ...−21,43,−81,163,..., the 121212th term of this sequence is....1211\frac{1}{2^{11}}21111212\frac{1}{2^{12}}21213211\frac{3}{2^{11}}21133212\frac{3}{2^{12}}21231211+1311\frac{1}{2^{11}} + \frac{1}{3^{11}}2111+3111ExplanationThe given sequence can be written as −121,322,−123,324,...-\frac{1}{2^1}, \frac{3}{2^2}, -\frac{1}{2^3}, \frac{3}{2^4}, ...−211,223,−231,243,... We can simplify it into a general form where For odd nnn, the nnnth term is un=−12nu_n = -\frac{1}{2^n}un=−2n1 For even nnn, the nnnth term is un=32nu_n = \frac{3}{2^n}un=2n3 Therefore, the 121212th term or even term is un=32nu_n = \frac{3}{2^n}un=2n3u12=3212u_{12} = \frac{3}{2^{12}}u12=2123
