题目

A sequence has terms a1,a2,a3,…….,an, where a1=2 and general term is given as an=a(n−1)× .

Quantity A

a6

Quantity B

214(a20)

选项

A.

Quantity A is greater.


B.

Quantity B is greater.


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。
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