The spell grew geometrically, as it rippled through the air.

咒语在空气中扩散，呈数增长着。

One approach involves adding up every chance the human has to win, using geometric series.

一种方法是通使用数，将人类获胜的每一种可能性相加。

The robot's total probability is also the total of a geometric series.

机器人的总概率也是一个数的总和。

We could bypass geometric series by starting with this reasoning.

我可以通以下推理来绕数。

Perhaps geometrically increased, and that's very unlikely to slow down.

或许呈数增长，而且这非常不可能会放缓。

Beyond that the difficulty of solving them rises in almost geometrical progression.

除此之外，解决的难度乎呈数上升。

They're called geometric series right, they're called geometric series.

被称为数，对，被称为数。

As one develops analytic listening skills, the opportunities to enjoy music increase geometrically.

随着分析聆听技能的提高，享受音乐的机会将呈数增加。

Two facts about geometric series are useful here.

关于数，有两个事实在这里很有用。

A geometric series is an infinite sum of numbers, where each number is the previous number multiplied by a common ratio.

数是指一个无限数列的和，其中每一项都是前一项乘以一个公比。

If we add all the possible probabilities of a human win, the total is the sum of a geometric series.

如果我把人类获胜的所有可能概率相加，总数就是数的和。

First, if the common ratio r of a geometric series has absolute value less than 1, the series has a finite total.

首先，如果数的公比r的绝对值小于1，那么这个数有一个有限的和。

Wildeve forgot the loss of the money at the sight of his lost love, whose preciousness in his eyes was increasing in geometrical progression with each new incident that reminded him of their hopeless division.

韦狄一见到他失去的爱人，顿时就把输钱一事忘了，每当新发生什么事都让他想起他之间使人绝望的分离，她的那份珍贵在他眼里，就以数增加。