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解答 - 几何数列

公比是：

r = 0.12010880415252545

r=0.12010880415252545

该系列的和是：

s = 44884

s=44884

此系列的通用形式是：

a_{n} = 40072 \cdot {0.12010880415252545}^{n - 1}

a_n=40072*0.12010880415252545^(n-1)

这个序列的第n项是：

40072, 4813, 578.083674386105, 69.43293883061298, 8.339507251740374, 1.0016482432278504, 0.12030677267557503, 0.014449902597513041, 0.001735560521107763, 0.0002084560987245873

40072,4813,578.083674386105,69.43293883061298,8.339507251740374,1.0016482432278504,0.12030677267557503,0.014449902597513041,0.001735560521107763,0.0002084560987245873

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：

$a_{2} a_{1} = \frac{4813}{40072} = 0.12010880415252545$

该序列的公比（ $r$ ）保持不变，并且等于两个连续项的商。
$r = 0.12010880415252545$

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： $a = 40, 072$ 、公比： $r = 0.12010880415252545$ 和元素数目 $n = 2$ 插入几何级数求和公式：

$s_{2} = 40072 * ((1 - {0.12010880415252545}^{2}) / (1 - 0.12010880415252545))$

$s_{2} = 40072 * ((1 - 0.014426124834949716) / (1 - 0.12010880415252545))$

$s_{2} = 40072 * (0.9855738751650502 / (1 - 0.12010880415252545))$

$s_{2} = 40072 * (0.9855738751650502 / 0.8798911958474745)$

$s_{2} = 40072 \cdot 1.1201088041525253$

$s_{2} = 44884.99999999999$

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： $a = 40, 072$ 和公比： $r = 0.12010880415252545$ 插入几何级数的公式：

$a_{n} = 40072 \cdot {0.12010880415252545}^{n - 1}$

4. 找到第n项

使用通用公式找到第n项

$a_{1} = 40072$

$a_{2} = a_{1} \cdot r^{n - 1} = 40072 \cdot {0.12010880415252545}^{2 - 1} = 40072 \cdot {0.12010880415252545}^{1} = 40072 \cdot 0.12010880415252545 = 4813$

$a_{3} = a_{1} \cdot r^{n - 1} = 40072 \cdot {0.12010880415252545}^{3 - 1} = 40072 \cdot {0.12010880415252545}^{2} = 40072 \cdot 0.014426124834949716 = 578.083674386105$

$a_{4} = a_{1} \cdot r^{n - 1} = 40072 \cdot {0.12010880415252545}^{4 - 1} = 40072 \cdot {0.12010880415252545}^{3} = 40072 \cdot 0.0017327046024808589 = 69.43293883061298$

$a_{5} = a_{1} \cdot r^{n - 1} = 40072 \cdot {0.12010880415252545}^{5 - 1} = 40072 \cdot {0.12010880415252545}^{4} = 40072 \cdot 0.00020811307775355295 = 8.339507251740374$

$a_{6} = a_{1} \cdot r^{n - 1} = 40072 \cdot {0.12010880415252545}^{6 - 1} = 40072 \cdot {0.12010880415252545}^{5} = 40072 \cdot 2.4996212897480792 E - 05 = 1.0016482432278504$

$a_{7} = a_{1} \cdot r^{n - 1} = 40072 \cdot {0.12010880415252545}^{7 - 1} = 40072 \cdot {0.12010880415252545}^{6} = 40072 \cdot 3.002265239458351 E - 06 = 0.12030677267557503$

$a_{8} = a_{1} \cdot r^{n - 1} = 40072 \cdot {0.12010880415252545}^{8 - 1} = 40072 \cdot {0.12010880415252545}^{7} = 40072 \cdot 3.60598487660038 E - 07 = 0.014449902597513041$

$a_{9} = a_{1} \cdot r^{n - 1} = 40072 \cdot {0.12010880415252545}^{9 - 1} = 40072 \cdot {0.12010880415252545}^{8} = 40072 \cdot 4.331105313205637 E - 08 = 0.001735560521107763$

$a_{10} = a_{1} \cdot r^{n - 1} = 40072 \cdot {0.12010880415252545}^{10 - 1} = 40072 \cdot {0.12010880415252545}^{9} = 40072 \cdot 5.202038798277783 E - 09 = 0.0002084560987245873$

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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