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解答 - 几何数列

公比是：

r = 0.3090909090909091

r=0.3090909090909091

该系列的和是：

s = 72

s=72

此系列的通用形式是：

a_{n} = 55 \cdot {0.3090909090909091}^{n - 1}

a_n=55*0.3090909090909091^(n-1)

这个序列的第n项是：

55, 17, 5.254545454545454, 1.6241322314049584, 0.5020045078888054, 0.15516502971108528, 0.04796010009251727, 0.014824030937687156, 0.004581973198921484, 0.0014162462614848224

55,17,5.254545454545454,1.6241322314049584,0.5020045078888054,0.15516502971108528,0.04796010009251727,0.014824030937687156,0.004581973198921484,0.0014162462614848224

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：

$a_{2} a_{1} = \frac{17}{55} = 0.3090909090909091$

该序列的公比（ $r$ ）保持不变，并且等于两个连续项的商。
$r = 0.3090909090909091$

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： $a = 55$ 、公比： $r = 0.3090909090909091$ 和元素数目 $n = 2$ 插入几何级数求和公式：

$s_{2} = 55 * ((1 - {0.3090909090909091}^{2}) / (1 - 0.3090909090909091))$

$s_{2} = 55 * ((1 - 0.09553719008264462) / (1 - 0.3090909090909091))$

$s_{2} = 55 * (0.9044628099173554 / (1 - 0.3090909090909091))$

$s_{2} = 55 * (0.9044628099173554 / 0.6909090909090909)$

$s_{2} = 55 \cdot 1.309090909090909$

$s_{2} = 72$

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： $a = 55$ 和公比： $r = 0.3090909090909091$ 插入几何级数的公式：

$a_{n} = 55 \cdot {0.3090909090909091}^{n - 1}$

4. 找到第n项

使用通用公式找到第n项

$a_{1} = 55$

$a_{2} = a_{1} \cdot r^{n - 1} = 55 \cdot {0.3090909090909091}^{2 - 1} = 55 \cdot {0.3090909090909091}^{1} = 55 \cdot 0.3090909090909091 = 17$

$a_{3} = a_{1} \cdot r^{n - 1} = 55 \cdot {0.3090909090909091}^{3 - 1} = 55 \cdot {0.3090909090909091}^{2} = 55 \cdot 0.09553719008264462 = 5.254545454545454$

$a_{4} = a_{1} \cdot r^{n - 1} = 55 \cdot {0.3090909090909091}^{4 - 1} = 55 \cdot {0.3090909090909091}^{3} = 55 \cdot 0.02952967693463561 = 1.6241322314049584$

$a_{5} = a_{1} \cdot r^{n - 1} = 55 \cdot {0.3090909090909091}^{5 - 1} = 55 \cdot {0.3090909090909091}^{4} = 55 \cdot 0.00912735468888737 = 0.5020045078888054$

$a_{6} = a_{1} \cdot r^{n - 1} = 55 \cdot {0.3090909090909091}^{6 - 1} = 55 \cdot {0.3090909090909091}^{5} = 55 \cdot 0.0028211823583833688 = 0.15516502971108528$

$a_{7} = a_{1} \cdot r^{n - 1} = 55 \cdot {0.3090909090909091}^{7 - 1} = 55 \cdot {0.3090909090909091}^{6} = 55 \cdot 0.0008720018198639503 = 0.04796010009251727$

$a_{8} = a_{1} \cdot r^{n - 1} = 55 \cdot {0.3090909090909091}^{8 - 1} = 55 \cdot {0.3090909090909091}^{7} = 55 \cdot 0.00026952783523067557 = 0.014824030937687156$

$a_{9} = a_{1} \cdot r^{n - 1} = 55 \cdot {0.3090909090909091}^{9 - 1} = 55 \cdot {0.3090909090909091}^{8} = 55 \cdot 8.330860361675426 E - 05 = 0.004581973198921484$

$a_{10} = a_{1} \cdot r^{n - 1} = 55 \cdot {0.3090909090909091}^{10 - 1} = 55 \cdot {0.3090909090909091}^{9} = 55 \cdot 2.574993202699677 E - 05 = 0.0014162462614848224$

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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