题目

Problem

The arithmetic series $S$ is given by

S=2+5+8+11+\cdots+254

Find

(a) the number of terms in the series,

(2)

(b) the sum of the series.

(2)

解答

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这是等差数列，首项 $a=2$ ，公差 $d=3$ 。把末项 $254$ 代入第 $n$ 项公式即可。

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For an arithmetic series,

\begin{align*} u_n=&\,a+(n-1)d. \end{align*}

Here $a=2$ , $d=3$ and $u_n=254$ , so

\begin{align*} 254=&\,2+3(n-1)\\ 252=&\,3(n-1)\\ 84=&\,n-1\\ n=&\,85. \end{align*}

Therefore, there are $85$ terms.

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已经知道项数后，可以用等差数列求和公式。这里用首项和末项最直接。

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Using

\begin{align*} S_n=&\,\frac{n}{2}(a+l), \end{align*}

we get

\begin{align*} S_{85} =&\,\frac{85}{2}(2+254)\\ =&\,\frac{85}{2}(256)\\ =&\,10880. \end{align*}

So the sum of the series is $10880$ .