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解答 - 几何数列

公比是：

r = 0.046620046620046623

r=0.046620046620046623

该系列的和是：

s = 449

s=449

此系列的通用形式是：

a_{n} = 429 \cdot {0.046620046620046623}^{n - 1}

a_n=429*0.046620046620046623^(n-1)

这个序列的第n项是：

429, 20, 0.9324009324009325, 0.043468574937106416, 0.0020265069900748914, 9.447585035314179 E - 05, 4.404468547932019 E - 06, 2.053365290411198 E - 07, 9.57279855669556 E - 09, 4.4628431499746203 E - 10

429,20,0.9324009324009325,0.043468574937106416,0.0020265069900748914,9.447585035314179E-05,4.404468547932019E-06,2.053365290411198E-07,9.57279855669556E-09,4.4628431499746203E-10

查看步骤

逐步解答

1. 找到公比

通过将序列中的任何项除以前一项来找到公比：

$a_{2} a_{1} = \frac{20}{429} = 0.046620046620046623$

该序列的公比（ $r$ ）保持不变，并且等于两个连续项的商。
$r = 0.046620046620046623$

2. 求和

5 个额外步骤

$s_{n} = a * ((1 - r^{n}) / (1 - r))$

要找到系列的和，将第一项： $a = 429$ 、公比： $r = 0.046620046620046623$ 和元素数目 $n = 2$ 插入几何级数求和公式：

$s_{2} = 429 * ((1 - {0.046620046620046623}^{2}) / (1 - 0.046620046620046623))$

$s_{2} = 429 * ((1 - 0.0021734287468553205) / (1 - 0.046620046620046623))$

$s_{2} = 429 * (0.9978265712531447 / (1 - 0.046620046620046623))$

$s_{2} = 429 * (0.9978265712531447 / 0.9533799533799534)$

$s_{2} = 429 \cdot 1.0466200466200466$

$s_{2} = 449$

3. 找到通用形式

$a_{n} = a \cdot r^{n - 1}$

要找到系列的通用形式，将第一项： $a = 429$ 和公比： $r = 0.046620046620046623$ 插入几何级数的公式：

$a_{n} = 429 \cdot {0.046620046620046623}^{n - 1}$

4. 找到第n项

使用通用公式找到第n项

$a_{1} = 429$

$a_{2} = a_{1} \cdot r^{n - 1} = 429 \cdot {0.046620046620046623}^{2 - 1} = 429 \cdot {0.046620046620046623}^{1} = 429 \cdot 0.046620046620046623 = 20$

$a_{3} = a_{1} \cdot r^{n - 1} = 429 \cdot {0.046620046620046623}^{3 - 1} = 429 \cdot {0.046620046620046623}^{2} = 429 \cdot 0.0021734287468553205 = 0.9324009324009325$

$a_{4} = a_{1} \cdot r^{n - 1} = 429 \cdot {0.046620046620046623}^{4 - 1} = 429 \cdot {0.046620046620046623}^{3} = 429 \cdot 0.00010132534950374456 = 0.043468574937106416$

$a_{5} = a_{1} \cdot r^{n - 1} = 429 \cdot {0.046620046620046623}^{5 - 1} = 429 \cdot {0.046620046620046623}^{4} = 429 \cdot 4.7237925176570895 E - 06 = 0.0020265069900748914$

$a_{6} = a_{1} \cdot r^{n - 1} = 429 \cdot {0.046620046620046623}^{6 - 1} = 429 \cdot {0.046620046620046623}^{5} = 429 \cdot 2.202234273966009 E - 07 = 9.447585035314179 E - 05$

$a_{7} = a_{1} \cdot r^{n - 1} = 429 \cdot {0.046620046620046623}^{7 - 1} = 429 \cdot {0.046620046620046623}^{6} = 429 \cdot 1.0266826452055988 E - 08 = 4.404468547932019 E - 06$

$a_{8} = a_{1} \cdot r^{n - 1} = 429 \cdot {0.046620046620046623}^{8 - 1} = 429 \cdot {0.046620046620046623}^{7} = 429 \cdot 4.78639927834778 E - 10 = 2.053365290411198 E - 07$

$a_{9} = a_{1} \cdot r^{n - 1} = 429 \cdot {0.046620046620046623}^{9 - 1} = 429 \cdot {0.046620046620046623}^{8} = 429 \cdot 2.2314215749873103 E - 11 = 9.57279855669556 E - 09$

$a_{10} = a_{1} \cdot r^{n - 1} = 429 \cdot {0.046620046620046623}^{10 - 1} = 429 \cdot {0.046620046620046623}^{9} = 429 \cdot 1.0402897785488626 E - 12 = 4.4628431499746203 E - 10$

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几何序列常用于解释数学、物理、工程、生物、经济、计算机科学、金融等领域的概念，因此它们是我们工具箱中非常有用的工具。例如，几何序列最常见的应用之一就是计算已经获得或未付的复利，这是与金融相关的最常见活动之一，可能意味着赚取或失去大量的金钱！其他应用包括但不仅限于计算概率、测算随时间变化的放射性以及设计建筑物。

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几何数列