找到无限几何级数的公比

在数学中，算术关注数字的研究，同时强调与它们相关的传统运算。这些运算符是加法、减法、乘法、除法、指数和根式。算术可以说是所有其他数学分支的基础。几何学、测量学、三角学等所有这些领域都使用基本的算术运算来进行各种数学观察、计算和结论。

系列

将给定序列的所有项加起来就形成了一系列相关序列。因此，序列基本上是一个序列，只是在其所有项之间有一个加法运算符。现在，由于一个系列是由一个序列本身制成的，它的模式与它的父序列完全相同是很自然的。一个系列是确定的还是不确定的，这也取决于相关的序列本身。

例子

• 1 + 2 + 3 + 4 + …. is an infinite series of all the natural numbers.
• 10 + 20 + 30 + 40 is a finite series of the first four multiples of 10.

无限几何系列

这样一个序列，其中每个后续项与其前一项具有恒定比率，称为几何级数。从系列的定义可以看出，构成系列的所有项都是相加的。无限系列是没有尽头的系列。这意味着在无限几何级数中，这些项彼此之间的比率是恒定的，并且没有尽头。需要注意的是，该比率是连续的，即在整个系列中是恒定的，称为共同比率。要记住的另一个重要方面是公比的绝对值必须大于零，即|r| < 0。

通常，具有第一项 a 和公比 r 的有限几何级数表示为：

a, ar, ar ² , ar3, ar ⁴ , ...ar ^{n – 2} , ar ^{n – 1} , ar ⁿ 。

然而，一个无限的几何级数将被描述如下，

a, a ^r , ar ² , ar ³ , ar ⁴ , ...ar ¹⁰⁰⁰⁰⁰ ,...

例如，系列 2 + 4 + 8 + 16 + 32 + ...。是一个无限几何级数，因为每个连续的项都是通过将前一项乘以 2 获得的，并且由于该级数没有结束，即不是有限的。

求无限几何级数的公比。

回答：

In simple language, the common ratio of a geometric series means the quantity which is multiplied by each preceding term in order to form the succeeding term. This means that every term after the first one is to be multiplied with a fixed quantity in order to form an infinite geometric series. It also implies that if one were to divide each succeeding term with the term immediately preceding it, the quotient so obtained would be the common ratio of the given series. This is because of the general rule of division in mathematics. Divisor multiplied by quotient yields the dividend. Here, the words divisor, quotient, and dividend pertain to the preceding term, common ratio, and successive terms respectively. The common ratio, r of a geometric series is calculated using the following formula,

r = $\frac{a_n}{a_{n-1}}$ 

Where n represents the succeeding term and n – 1 is the term immediately preceding the term chosen.

It is to be noted that in order to calculate the common ratio, any random term can be picked up from the given series and divided by its immediately preceding term, and the quotient would be constant throughout the whole series, hence the name ‘common ratio’.

Examples

• The common ratio of the series 3, 9, 27, 81, 243, 729, ….. is = 9/3 = 27/9 = 81/27 = 243/81 = 729/243 = 3.
• The common ratio of the geometric series 10, 100, 1000, 10000, … would be = 100/10 = 1000/100 = 10.
• The common ratio of the series 56, 28, 14, 7,… would be = 56/2 = 28/14 = 14/7 = 2.

类似问题

问题 1：求级数的公比：27, 9, 3, 1, 1/3, 1/9, 1/27, ... 并列出接下来的三个项。

解决方案：

Common ratio = $\frac{a_n}{a_{n-1}} = \frac{\frac{1}{3}}{1}$ = 1/3.

The next three terms would be,

1/27 × 1/3 = 1/81

1/81 × 1/3 = 1/243

1/243 × 1/3 = 1/729

Hence the common ratio is 1/3 and the next three terms are 1/81, 1/243, 1/729.

问题 2：找出系列的公比：5、25、125、625，……并列出接下来的两项。

解决方案：

Common ratio = $\frac{a_n}{a_{n-1}}$ = 25/5 = 5

The next three terms would be,

625 × 5 = 3125

3125 × 5 = 15625

Hence the common ratio is 5 and the next two terms are 3125 and 15625.

问题 3. 找出系列的公比：25, 5, 1, 1/5, 1/25, ... 并列出接下来的 4 个项。

解决方案：

Common ratio = $\frac{a_n}{a_{n-1}}$ = 5/25 = 1/5

The next 4 terms would be,

1/25 × 1/5 = 1/125

1/125 × 1/5 = 1/625

1/625 × 1/5 = 1/3125

1/3125 × 1/5 = 1/15625

Hence the common ratio is 1/5 and the next 4 terms are 1/125, 1/625, 1/3125 and 1/15625. 

问题 4：找出系列的公比：1/216, 1/36, 1/6, 1, 6, 36, ... 并列出接下来的 2 个项。

解决方案：

Common ratio = $\frac{a_n}{a_{n-1}}$ = $\frac{\frac{1}{216}}{\frac{1}{36}}$ = 6

The next 2 terms would be,

36 × 6 = 216

216 × 6 = 1296

Hence the common ratio is 6 and the next 2 terms are 216 and 1296.

问题 5：找出系列的公比：1, 3, 9, 27, 81, ... 并列出接下来的 4 个项。

解决方案：

Common ratio = $\frac{a_n}{a_{n-1}}$ = 3/1 = 3

The next 2 terms would be:

81 × 3 = 243

243 × 3 = 729

Hence the common ratio is 3 and the next 2 terms are 243 and 729.