题目

Problem

The third term of a geometric progression is 18 and the sum of the first three terms is 26. It is given that the common ratio is negative.

(a) Find the tenth term of the progression. Give your answer correct to 3 significant figures.

[5]

(b) Find the exact value of the sum to infinity of the progression.

[2]

题目中文翻译

一个等比数列的第三项为 18，前三项之和为 26。已知公比为负。

(a) 求该数列的第 10 项。答案精确到 3 位有效数字。

[5]

(b) 求该数列无穷和的精确值。

[2]

解答

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第三项已经给出，所以前两项的和是 $8$ 。用公比把前两项表示成第三项的倍数。

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Let the first term be $a$ and the common ratio be $r<0$ . Since $ar^2=18$ and $a+ar=26-18=8$ ,

\frac{18}{r^2}(1+r)=8 \quad\Longrightarrow\quad 4r^2-9r-9=(4r+3)(r-3)=0.

The negative root is $r=-\frac34$ , giving $a=18/r^2=32$ . The tenth term is

32\Big(-\frac34\Big)^9=-2.40

to three significant figures.

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沿用 (a) 的精确首项和公比。虽然公比为负，但它的绝对值小于 $1$ 。

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Since $|r|=\frac34<1$ , the sum to infinity exists and is

S_\infty=\frac{32}{1-(-3/4)}=\frac{128}{7}.