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解答 - 几何数列

公比是：

r = 0.5185185185185185

r=0.5185185185185185

该系列的和是：

s = 123

s=123

此系列的通用形式是：

a_{n} = 81 \cdot {0.5185185185185185}^{n - 1}

a_n=81*0.5185185185185185^(n-1)

这个序列的第n项是：

81, 42, 21.777777777777775, 11.292181069958845, 5.8552049992379205, 3.036032221827069, 1.574238929836258, 0.8162720376928745, 0.4232521676926016, 0.21946408695171932

81,42,21.777777777777775,11.292181069958845,5.8552049992379205,3.036032221827069,1.574238929836258,0.8162720376928745,0.4232521676926016,0.21946408695171932

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：

$a_{2} a_{1} = \frac{42}{81} = 0.5185185185185185$

该序列的公比（ $r$ ）保持不变，并且等于两个连续项的商。
$r = 0.5185185185185185$

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： $r = 0.5185185185185185$ 和元素数目 $n = 2$ 插入几何级数求和公式：

$s_{2} = 81 * ((1 - {0.5185185185185185}^{2}) / (1 - 0.5185185185185185))$

$s_{2} = 81 * ((1 - 0.2688614540466392) / (1 - 0.5185185185185185))$

$s_{2} = 81 * (0.7311385459533608 / (1 - 0.5185185185185185))$

$s_{2} = 81 * (0.7311385459533608 / 0.4814814814814815)$

$s_{2} = 81 \cdot 1.5185185185185186$

$s_{2} = 123$

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： $r = 0.5185185185185185$ 插入几何级数的公式：

$a_{n} = 81 \cdot {0.5185185185185185}^{n - 1}$

4. 找到第n项

使用通用公式找到第n项

$a_{1} = 81$

$a_{2} = a_{1} \cdot r^{n - 1} = 81 \cdot {0.5185185185185185}^{2 - 1} = 81 \cdot {0.5185185185185185}^{1} = 81 \cdot 0.5185185185185185 = 42$

$a_{3} = a_{1} \cdot r^{n - 1} = 81 \cdot {0.5185185185185185}^{3 - 1} = 81 \cdot {0.5185185185185185}^{2} = 81 \cdot 0.2688614540466392 = 21.777777777777775$

$a_{4} = a_{1} \cdot r^{n - 1} = 81 \cdot {0.5185185185185185}^{4 - 1} = 81 \cdot {0.5185185185185185}^{3} = 81 \cdot 0.1394096428389981 = 11.292181069958845$

$a_{5} = a_{1} \cdot r^{n - 1} = 81 \cdot {0.5185185185185185}^{5 - 1} = 81 \cdot {0.5185185185185185}^{4} = 81 \cdot 0.07228648147207309 = 5.8552049992379205$

$a_{6} = a_{1} \cdot r^{n - 1} = 81 \cdot {0.5185185185185185}^{6 - 1} = 81 \cdot {0.5185185185185185}^{5} = 81 \cdot 0.03748187928181567 = 3.036032221827069$

$a_{7} = a_{1} \cdot r^{n - 1} = 81 \cdot {0.5185185185185185}^{7 - 1} = 81 \cdot {0.5185185185185185}^{6} = 81 \cdot 0.01943504851649701 = 1.574238929836258$

$a_{8} = a_{1} \cdot r^{n - 1} = 81 \cdot {0.5185185185185185}^{8 - 1} = 81 \cdot {0.5185185185185185}^{7} = 81 \cdot 0.010077432564109562 = 0.8162720376928745$

$a_{9} = a_{1} \cdot r^{n - 1} = 81 \cdot {0.5185185185185185}^{9 - 1} = 81 \cdot {0.5185185185185185}^{8} = 81 \cdot 0.005225335403612365 = 0.4232521676926016$

$a_{10} = a_{1} \cdot r^{n - 1} = 81 \cdot {0.5185185185185185}^{10 - 1} = 81 \cdot {0.5185185185185185}^{9} = 81 \cdot 0.0027094331722434485 = 0.21946408695171932$

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几何数列