➕ 数列求和：从 1 加到 100 的聪明办法➕ Series Summation: the clever way to add 1 to 100
上一片里，荷叶翻倍得又快又猛。可要是把数列的每一项都加起来——比如从 1 加到 100——总不能老老实实加一百次吧？这一片找一条捷径。Last leaf, lily pads doubled fast and furious. But add up every term — say 1 through 100 — would you honestly add a hundred times? This leaf finds the shortcut.
第 1 步：首尾配对Step 1: Pair the two ends
先看一个最简单的问题：把 1 到 100 排成一排，最小的数和最大的数放在一起，会发生什么？Start with the simplest question: line up 1 to 100, put the smallest next to the largest — what happens?
1 配 100、2 配 99、3 配 98……每一对加起来都是 101。这不是巧合：1+100 之后，小的每 +1，大的就 −1，和一直不变。像这样把首尾两项配成一对，叫首尾配对。1 pairs with 100, 2 with 99, 3 with 98… every pair adds up to 101. No coincidence: after 1+100, each time the small number grows by 1 the large one shrinks by 1 — the sum never changes. Pairing the two ends like this is called end-pairing.
每对都是 101，那到底能配出多少对呢？Every pair makes 101 — but how many pairs are there?
第 2 步：数一数有多少对Step 2: Count the pairs
100 个数，两两配对，正好 50 对。50 × 101 = 5050：加法变成乘法，一步出答案。传说高斯小时候就是这样做完老师出的「从 1 加到 100」的。A hundred numbers, two to a pair — exactly 50 pairs. 50 × 101 = 5050: addition turns into multiplication and the answer lands in one step. Legend says young Gauss finished his teacher's “add 1 to 100” chore exactly this way.
如果数列不是 1 到 100，而是 1 到 n 呢？把数字摆成图形，公式自己会浮出来。What if the sequence is 1 to n instead of 1 to 100? Lay the numbers out as a picture — the formula surfaces by itself.
第 3 步：摆成梯形，一眼看穿Step 3: Stack it into a staircase
把 1, 2, 3, …, n 用方块摞成阶梯，复制一份倒过来一拼，就是一个 n × (n+1) 的矩形，阶梯正好占一半：所以 1+2+…+n = n(n+1)/2。拿 n=10 验证：(1+10) × 10 ÷ 2 = 55。等比数列求和也有自己的招：把整个和乘 q 再错位相减——比如 1+2+4+…+512，乘 2 得 2+4+…+1024，两式相减中间全消掉，只剩 1024 − 1 = 1023。Stack 1, 2, 3, …, n into a staircase of unit squares, then put a flipped copy beside it: together they form an n × (n+1) rectangle, and the staircase is exactly half — so 1+2+…+n = n(n+1)/2. Check with n = 10: (1+10) × 10 ÷ 2 = 55. Geometric sums have their own trick: multiply the whole sum by q and subtract — take 1+2+4+…+512, multiply by 2 to get 2+4+…+1024, subtract, and everything in between cancels: 1024 − 1 = 1023.
🎮 你来配对（1 分钟）🎮 Your turn: pair them up (1 minute)
道理讲完了。拖动卡片，把左边和右边配成 50 对里的 8 对——每对都得凑成 101，配完看看总和怎么算。Theory done. Drag the cards and pair up 8 of the 50 pairs — every pair must make 101, then watch the total fall into place.
一句话记住它：首尾配对，每对的和都一样——数对数、做乘法，加法题秒变乘法题。Remember it in one line: pair the ends and every pair shares the same sum — count the pairs, multiply, and a long addition becomes one multiplication.
首尾配对：1+100 = 2+99 = … = 101，和一直不变End-pairing: 1+100 = 2+99 = … = 101 — the sum never changes50 对 × 101 = 5050：从 1 加到 100 一步完成50 pairs × 101 = 5050: adding 1 to 100 in one step通用公式 1+2+…+n = n(n+1)/2；等比求和用「乘 q 错位相减」General formula 1+2+…+n = n(n+1)/2; geometric sums use multiply-by-q and subtract
内容参考 OpenStax 等公开教材，多来源核对 · AI 生成、人工审核 · 发现错误欢迎指正，帮这片叶子长得更好。Based on OpenStax and other open textbooks, cross-checked across sources · AI-generated, human-reviewed · Spotted a mistake? Tell us — help this leaf grow.
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