题目
A sequence has terms a1,a2,a3,…….,an, where a1=2 and general term is given as an=a(n−1)× .
选项
C.The two quantities are equal.
D.The relationship cannot be determined from the information given.
解析
首先,我们计算几项来找出规律:
$a_2 = a_1 \times \frac{1}{2} = 2 \times \frac{1}{2} = 1 = 2^0$
$a_3 = a_2 \times \frac{1}{2} = 2 \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{2} = 2^{-1}$
可以看出,数列的第 $n$ 项也可以表示为:
$a_n = \frac{1}{2^{n-1}}$
因此,A 列:
$a_6 = \frac{1}{2^{6-1}} = \frac{1}{2^5}$
B 列可化简为:
$2^{14}(a_{20}) = 2^{14} \times \left( \frac{1}{2^{20-1}} \right) = 2^{14} \times \frac{1}{2^{19}} = \frac{1}{2^{19-14}} = \frac{1}{2^5}$
A 列与 B 列的值相等。