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解答 - 几何数列

公比是：

r = 0.6538461538461539

r=0.6538461538461539

该系列的和是：

s = 129

s=129

此系列的通用形式是：

a_{n} = 78 \cdot {0.6538461538461539}^{n - 1}

a_n=78*0.6538461538461539^(n-1)

这个序列的第n项是：

78, 51, 33.34615384615385, 21.803254437869825, 14.255974055530269, 9.321213805539022, 6.0946397959293614, 3.9849567896461204, 2.6055486701532327, 1.7036279766386522

78,51,33.34615384615385,21.803254437869825,14.255974055530269,9.321213805539022,6.0946397959293614,3.9849567896461204,2.6055486701532327,1.7036279766386522

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：

$a_{2} a_{1} = \frac{51}{78} = 0.6538461538461539$

该序列的公比（ $r$ ）保持不变，并且等于两个连续项的商。
$r = 0.6538461538461539$

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： $a = 78$ 、公比： $r = 0.6538461538461539$ 和元素数目 $n = 2$ 插入几何级数求和公式：

$s_{2} = 78 * ((1 - {0.6538461538461539}^{2}) / (1 - 0.6538461538461539))$

$s_{2} = 78 * ((1 - 0.4275147928994083) / (1 - 0.6538461538461539))$

$s_{2} = 78 * (0.5724852071005917 / (1 - 0.6538461538461539))$

$s_{2} = 78 * (0.5724852071005917 / 0.34615384615384615)$

$s_{2} = 78 \cdot 1.6538461538461537$

$s_{2} = 129$

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： $a = 78$ 和公比： $r = 0.6538461538461539$ 插入几何级数的公式：

$a_{n} = 78 \cdot {0.6538461538461539}^{n - 1}$

4. 找到第n项

使用通用公式找到第n项

$a_{1} = 78$

$a_{2} = a_{1} \cdot r^{n - 1} = 78 \cdot {0.6538461538461539}^{2 - 1} = 78 \cdot {0.6538461538461539}^{1} = 78 \cdot 0.6538461538461539 = 51$

$a_{3} = a_{1} \cdot r^{n - 1} = 78 \cdot {0.6538461538461539}^{3 - 1} = 78 \cdot {0.6538461538461539}^{2} = 78 \cdot 0.4275147928994083 = 33.34615384615385$

$a_{4} = a_{1} \cdot r^{n - 1} = 78 \cdot {0.6538461538461539}^{4 - 1} = 78 \cdot {0.6538461538461539}^{3} = 78 \cdot 0.27952890304961314 = 21.803254437869825$

$a_{5} = a_{1} \cdot r^{n - 1} = 78 \cdot {0.6538461538461539}^{5 - 1} = 78 \cdot {0.6538461538461539}^{4} = 78 \cdot 0.18276889814782396 = 14.255974055530269$

$a_{6} = a_{1} \cdot r^{n - 1} = 78 \cdot {0.6538461538461539}^{6 - 1} = 78 \cdot {0.6538461538461539}^{5} = 78 \cdot 0.11950274109665414 = 9.321213805539022$

$a_{7} = a_{1} \cdot r^{n - 1} = 78 \cdot {0.6538461538461539}^{7 - 1} = 78 \cdot {0.6538461538461539}^{6} = 78 \cdot 0.07813640764012002 = 6.0946397959293614$

$a_{8} = a_{1} \cdot r^{n - 1} = 78 \cdot {0.6538461538461539}^{8 - 1} = 78 \cdot {0.6538461538461539}^{7} = 78 \cdot 0.0510891896108477 = 3.9849567896461204$

$a_{9} = a_{1} \cdot r^{n - 1} = 78 \cdot {0.6538461538461539}^{9 - 1} = 78 \cdot {0.6538461538461539}^{8} = 78 \cdot 0.03340447013016965 = 2.6055486701532327$

$a_{10} = a_{1} \cdot r^{n - 1} = 78 \cdot {0.6538461538461539}^{10 - 1} = 78 \cdot {0.6538461538461539}^{9} = 78 \cdot 0.021841384315880157 = 1.7036279766386522$

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