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解答 - 几何数列

公比是：

r = 0.07407407407407407

r=0.07407407407407407

该系列的和是：

s = 57

s=57

此系列的通用形式是：

a_{n} = 54 \cdot {0.07407407407407407}^{n - 1}

a_n=54*0.07407407407407407^(n-1)

这个序列的第n项是：

54, 4, 0.2962962962962963, 0.021947873799725647, 0.0016257684296093072, 0.0001204272910821709, 8.920540080160806 E - 06, 6.607807466785782 E - 07, 4.8946721976190975 E - 08, 3.62568310934748 E - 09

54,4,0.2962962962962963,0.021947873799725647,0.0016257684296093072,0.0001204272910821709,8.920540080160806E-06,6.607807466785782E-07,4.8946721976190975E-08,3.62568310934748E-09

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：

$a_{2} a_{1} = \frac{4}{54} = 0.07407407407407407$

该序列的公比（ $r$ ）保持不变，并且等于两个连续项的商。
$r = 0.07407407407407407$

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： $a = 54$ 、公比： $r = 0.07407407407407407$ 和元素数目 $n = 2$ 插入几何级数求和公式：

$s_{2} = 54 * ((1 - {0.07407407407407407}^{2}) / (1 - 0.07407407407407407))$

$s_{2} = 54 * ((1 - 0.0054869684499314125) / (1 - 0.07407407407407407))$

$s_{2} = 54 * (0.9945130315500685 / (1 - 0.07407407407407407))$

$s_{2} = 54 * (0.9945130315500685 / 0.9259259259259259)$

$s_{2} = 54 \cdot 1.074074074074074$

$s_{2} = 57.99999999999999$

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： $a = 54$ 和公比： $r = 0.07407407407407407$ 插入几何级数的公式：

$a_{n} = 54 \cdot {0.07407407407407407}^{n - 1}$

4. 找到第n项

使用通用公式找到第n项

$a_{1} = 54$

$a_{2} = a_{1} \cdot r^{n - 1} = 54 \cdot {0.07407407407407407}^{2 - 1} = 54 \cdot {0.07407407407407407}^{1} = 54 \cdot 0.07407407407407407 = 4$

$a_{3} = a_{1} \cdot r^{n - 1} = 54 \cdot {0.07407407407407407}^{3 - 1} = 54 \cdot {0.07407407407407407}^{2} = 54 \cdot 0.0054869684499314125 = 0.2962962962962963$

$a_{4} = a_{1} \cdot r^{n - 1} = 54 \cdot {0.07407407407407407}^{4 - 1} = 54 \cdot {0.07407407407407407}^{3} = 54 \cdot 0.0004064421074023268 = 0.021947873799725647$

$a_{5} = a_{1} \cdot r^{n - 1} = 54 \cdot {0.07407407407407407}^{5 - 1} = 54 \cdot {0.07407407407407407}^{4} = 54 \cdot 3.0106822770542726 E - 05 = 0.0016257684296093072$

$a_{6} = a_{1} \cdot r^{n - 1} = 54 \cdot {0.07407407407407407}^{6 - 1} = 54 \cdot {0.07407407407407407}^{5} = 54 \cdot 2.230135020040202 E - 06 = 0.0001204272910821709$

$a_{7} = a_{1} \cdot r^{n - 1} = 54 \cdot {0.07407407407407407}^{7 - 1} = 54 \cdot {0.07407407407407407}^{6} = 54 \cdot 1.6519518666964456 E - 07 = 8.920540080160806 E - 06$

$a_{8} = a_{1} \cdot r^{n - 1} = 54 \cdot {0.07407407407407407}^{8 - 1} = 54 \cdot {0.07407407407407407}^{7} = 54 \cdot 1.2236680494047745 E - 08 = 6.607807466785782 E - 07$

$a_{9} = a_{1} \cdot r^{n - 1} = 54 \cdot {0.07407407407407407}^{9 - 1} = 54 \cdot {0.07407407407407407}^{8} = 54 \cdot 9.064207773368699 E - 10 = 4.8946721976190975 E - 08$

$a_{10} = a_{1} \cdot r^{n - 1} = 54 \cdot {0.07407407407407407}^{10 - 1} = 54 \cdot {0.07407407407407407}^{9} = 54 \cdot 6.714227980273111 E - 11 = 3.62568310934748 E - 09$

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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