Question

1-1/2+1/2^2-1/2^3+ … +1/2^9

Answer 1

$1 - \frac{1}{ {2}} + \frac{1}{ {2}^{2} } - \frac{1}{ {2}^{3} } + ... - \frac{1}{ {2}^{9} } = \frac{ {2}^{9} - {2}^{8} + {2}^{7} - ... - {2}^{0} }{ {2}^{9} }$

calculam numaratorul

$s = {2}^{9} - {2}^{8} + {2}^{7} - .... - 1$

grupa 2 cate 2(2 la a 9 a cu 2 la a 8 a, 2 la a 7 a cu 2 la 6 a si tot asa)

si obtinem

$s = 2 + 2 + 2 + .. + 2 = 2 \times 5 = 10$

inlocuim si obtinem

$\frac{ {2}^{9} - {2}^{8} + {2}^{7} - ... - {2}^{0} }{ {2}^{9} } = \frac{10}{ {2}^{8} } = \frac{5}{ {2}^{8} } = \frac{5}{256}$