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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 = 86

s=86

此系列的通用形式是：

a_{n} = 81 \cdot {0.07407407407407407}^{n - 1}

a_n=81*0.07407407407407407^(n-1)

这个序列的第n项是：

81, 6, 0.4444444444444444, 0.03292181069958847, 0.002438652644413961, 0.00018064093662325636, 1.3380810120241209 E - 05, 9.911711200178673 E - 07, 7.342008296428646 E - 08, 5.43852466402122 E - 09

81,6,0.4444444444444444,0.03292181069958847,0.002438652644413961,0.00018064093662325636,1.3380810120241209E-05,9.911711200178673E-07,7.342008296428646E-08,5.43852466402122E-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：

$a_{2} a_{1} = \frac{6}{81} = 0.07407407407407407$

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： $a = 81$ 、公比： $r = 0.07407407407407407$ 和元素数目 $n = 2$ 插入几何级数求和公式：

$s_{2} = 81 * ((1 - {0.07407407407407407}^{2}) / (1 - 0.07407407407407407))$

$s_{2} = 81 * ((1 - 0.0054869684499314125) / (1 - 0.07407407407407407))$

$s_{2} = 81 * (0.9945130315500685 / (1 - 0.07407407407407407))$

$s_{2} = 81 * (0.9945130315500685 / 0.9259259259259259)$

$s_{2} = 81 \cdot 1.074074074074074$

$s_{2} = 86.99999999999999$

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： $a = 81$ 和公比： $r = 0.07407407407407407$ 插入几何级数的公式：

$a_{n} = 81 \cdot {0.07407407407407407}^{n - 1}$

4. 找到第n项

使用通用公式找到第n项

$a_{1} = 81$

$a_{2} = a_{1} \cdot r^{n - 1} = 81 \cdot {0.07407407407407407}^{2 - 1} = 81 \cdot {0.07407407407407407}^{1} = 81 \cdot 0.07407407407407407 = 6$

$a_{3} = a_{1} \cdot r^{n - 1} = 81 \cdot {0.07407407407407407}^{3 - 1} = 81 \cdot {0.07407407407407407}^{2} = 81 \cdot 0.0054869684499314125 = 0.4444444444444444$

$a_{4} = a_{1} \cdot r^{n - 1} = 81 \cdot {0.07407407407407407}^{4 - 1} = 81 \cdot {0.07407407407407407}^{3} = 81 \cdot 0.0004064421074023268 = 0.03292181069958847$

$a_{5} = a_{1} \cdot r^{n - 1} = 81 \cdot {0.07407407407407407}^{5 - 1} = 81 \cdot {0.07407407407407407}^{4} = 81 \cdot 3.0106822770542726 E - 05 = 0.002438652644413961$

$a_{6} = a_{1} \cdot r^{n - 1} = 81 \cdot {0.07407407407407407}^{6 - 1} = 81 \cdot {0.07407407407407407}^{5} = 81 \cdot 2.230135020040202 E - 06 = 0.00018064093662325636$

$a_{7} = a_{1} \cdot r^{n - 1} = 81 \cdot {0.07407407407407407}^{7 - 1} = 81 \cdot {0.07407407407407407}^{6} = 81 \cdot 1.6519518666964456 E - 07 = 1.3380810120241209 E - 05$

$a_{8} = a_{1} \cdot r^{n - 1} = 81 \cdot {0.07407407407407407}^{8 - 1} = 81 \cdot {0.07407407407407407}^{7} = 81 \cdot 1.2236680494047745 E - 08 = 9.911711200178673 E - 07$

$a_{9} = a_{1} \cdot r^{n - 1} = 81 \cdot {0.07407407407407407}^{9 - 1} = 81 \cdot {0.07407407407407407}^{8} = 81 \cdot 9.064207773368699 E - 10 = 7.342008296428646 E - 08$

$a_{10} = a_{1} \cdot r^{n - 1} = 81 \cdot {0.07407407407407407}^{10 - 1} = 81 \cdot {0.07407407407407407}^{9} = 81 \cdot 6.714227980273111 E - 11 = 5.43852466402122 E - 09$

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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