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解答 - 几何数列

公比是：

r = 0.7094035698737484

r=0.7094035698737484

该系列的和是：

s = 7853

s=7853

此系列的通用形式是：

a_{n} = 4594 \cdot {0.7094035698737484}^{n - 1}

a_n=4594*0.7094035698737484^(n-1)

这个序列的第n项是：

4594, 3259.0000000000005, 2311.946234218546, 1640.102911910806, 1163.4948606698556, 825.3874076889551, 585.5327735433839, 415.3790398297536, 294.6713737059571, 209.0409244466074

4594,3259.0000000000005,2311.946234218546,1640.102911910806,1163.4948606698556,825.3874076889551,585.5327735433839,415.3790398297536,294.6713737059571,209.0409244466074

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：

$a_{2} a_{1} = \frac{3259}{4594} = 0.7094035698737484$

该序列的公比（ $r$ ）保持不变，并且等于两个连续项的商。
$r = 0.7094035698737484$

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： $a = 4, 594$ 、公比： $r = 0.7094035698737484$ 和元素数目 $n = 2$ 插入几何级数求和公式：

$s_{2} = 4594 * ((1 - {0.7094035698737484}^{2}) / (1 - 0.7094035698737484))$

$s_{2} = 4594 * ((1 - 0.5032534249496182) / (1 - 0.7094035698737484))$

$s_{2} = 4594 * (0.4967465750503818 / (1 - 0.7094035698737484))$

$s_{2} = 4594 * (0.4967465750503818 / 0.2905964301262516)$

$s_{2} = 4594 \cdot 1.7094035698737486$

$s_{2} = 7853.000000000001$

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： $a = 4, 594$ 和公比： $r = 0.7094035698737484$ 插入几何级数的公式：

$a_{n} = 4594 \cdot {0.7094035698737484}^{n - 1}$

4. 找到第n项

使用通用公式找到第n项

$a_{1} = 4594$

$a_{2} = a_{1} \cdot r^{n - 1} = 4594 \cdot {0.7094035698737484}^{2 - 1} = 4594 \cdot {0.7094035698737484}^{1} = 4594 \cdot 0.7094035698737484 = 3259.0000000000005$

$a_{3} = a_{1} \cdot r^{n - 1} = 4594 \cdot {0.7094035698737484}^{3 - 1} = 4594 \cdot {0.7094035698737484}^{2} = 4594 \cdot 0.5032534249496182 = 2311.946234218546$

$a_{4} = a_{1} \cdot r^{n - 1} = 4594 \cdot {0.7094035698737484}^{4 - 1} = 4594 \cdot {0.7094035698737484}^{3} = 4594 \cdot 0.3570097762104497 = 1640.102911910806$

$a_{5} = a_{1} \cdot r^{n - 1} = 4594 \cdot {0.7094035698737484}^{5 - 1} = 4594 \cdot {0.7094035698737484}^{4} = 4594 \cdot 0.25326400972352103 = 1163.4948606698556$

$a_{6} = a_{1} \cdot r^{n - 1} = 4594 \cdot {0.7094035698737484}^{6 - 1} = 4594 \cdot {0.7094035698737484}^{5} = 4594 \cdot 0.17966639261840556 = 825.3874076889551$

$a_{7} = a_{1} \cdot r^{n - 1} = 4594 \cdot {0.7094035698737484}^{7 - 1} = 4594 \cdot {0.7094035698737484}^{6} = 4594 \cdot 0.1274559803098354 = 585.5327735433839$

$a_{8} = a_{1} \cdot r^{n - 1} = 4594 \cdot {0.7094035698737484}^{8 - 1} = 4594 \cdot {0.7094035698737484}^{7} = 4594 \cdot 0.09041772743355542 = 415.3790398297536$

$a_{9} = a_{1} \cdot r^{n - 1} = 4594 \cdot {0.7094035698737484}^{9 - 1} = 4594 \cdot {0.7094035698737484}^{8} = 4594 \cdot 0.06414265862123576 = 294.6713737059571$

$a_{10} = a_{1} \cdot r^{n - 1} = 4594 \cdot {0.7094035698737484}^{10 - 1} = 4594 \cdot {0.7094035698737484}^{9} = 4594 \cdot 0.04550303100709782 = 209.0409244466074$

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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