题目

Problem

The first term of an arithmetic progression is $-20$ and the common difference is $5$ .

(a) Find the sum of the first $20$ terms of the progression.

[2]

It is given that the sum of the first $2k$ terms is $10$ times the sum of the first $k$ terms.

(b) Find the value of $k$ .

[3]

题目中文翻译

一个等差数列的首项为 $-20$ ，公差为 $5$ 。

(a) 求该数列前 $20$ 项的和。

已知前 $2k$ 项的和是前 $k$ 项和的 $10$ 倍。

(b) 求 $k$ 的值。

解答

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用等差数列求和公式，把首项、公差和项数代入。

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The sum is

S_{20}=\frac{20}{2}\bigl(2(-20)+19\cdot5\bigr)=550.

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用同一个求和公式写出 $S_k$ 和 $S_{2k}$ ，再按题意令前者的十倍等于后者。

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The sums are

S_k=\frac k2(5k-45),\qquad S_{2k}=k(10k-45).

Since $S_{2k}=10S_k$ and $k>0$ , division by $k$ gives $10k-45=5(5k-45)$ . Hence $15k=180$ and $k=12$ .