AMC10 2003 B

AMC10 2003 B · Q8

AMC10 2003 B · Q8. It mainly tests Sequences & recursion (algebra).

The second and fourth terms of a geometric sequence are 2 and 6. Which of the following is a possible first term?

一个几何数列的第二项和第四项分别为 2 和 6。下列哪项可能是首项？

(A) -√3 -√3

(B) -2√3/3 -2√3/3

(D) √3 √3

(E) 3 3

Answer

Correct choice: (B)

正确答案：（B）

Solution

(B) Let the sequence be denoted $a, ar, ar^2, ar^3, \ldots$, with $ar = 2$ and $ar^3 = 6$. Then $r^2 = 3$ and $r = \sqrt{3}$ or $r = -\sqrt{3}$. Therefore $a = \frac{2\sqrt{3}}{3}$ or $a = -\frac{2\sqrt{3}}{3}$.

（B）设数列表示为 $a, ar, ar^2, ar^3, \ldots$，且 $ar = 2$、$ar^3 = 6$。则 $r^2 = 3$，所以 $r = \sqrt{3}$ 或 $r = -\sqrt{3}$。因此 $a = \frac{2\sqrt{3}}{3}$ 或 $a = -\frac{2\sqrt{3}}{3}$。

Topics

AL.012 Sequences & recursion (algebra)