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解答 - 几何数列

公比是：

r = 0.2727272727272727

r=0.2727272727272727

该系列的和是：

s = 42

s=42

此系列的通用形式是：

a_{n} = 33 \cdot {0.2727272727272727}^{n - 1}

a_n=33*0.2727272727272727^(n-1)

这个序列的第n项是：

33, 9, 2.454545454545454, 0.6694214876033056, 0.18256949661908337, 0.04979168089611363, 0.013579549335303717, 0.003703513455082832, 0.0010100491241134995, 0.0002754679429400453

33,9,2.454545454545454,0.6694214876033056,0.18256949661908337,0.04979168089611363,0.013579549335303717,0.003703513455082832,0.0010100491241134995,0.0002754679429400453

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：

$a_{2} a_{1} = \frac{9}{33} = 0.2727272727272727$

该序列的公比（ $r$ ）保持不变，并且等于两个连续项的商。
$r = 0.2727272727272727$

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： $a = 33$ 、公比： $r = 0.2727272727272727$ 和元素数目 $n = 2$ 插入几何级数求和公式：

$s_{2} = 33 * ((1 - {0.2727272727272727}^{2}) / (1 - 0.2727272727272727))$

$s_{2} = 33 * ((1 - 0.07438016528925619) / (1 - 0.2727272727272727))$

$s_{2} = 33 * (0.9256198347107438 / (1 - 0.2727272727272727))$

$s_{2} = 33 * (0.9256198347107438 / 0.7272727272727273)$

$s_{2} = 33 \cdot 1.2727272727272727$

$s_{2} = 42$

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： $a = 33$ 和公比： $r = 0.2727272727272727$ 插入几何级数的公式：

$a_{n} = 33 \cdot {0.2727272727272727}^{n - 1}$

4. 找到第n项

使用通用公式找到第n项

$a_{1} = 33$

$a_{2} = a_{1} \cdot r^{n - 1} = 33 \cdot {0.2727272727272727}^{2 - 1} = 33 \cdot {0.2727272727272727}^{1} = 33 \cdot 0.2727272727272727 = 9$

$a_{3} = a_{1} \cdot r^{n - 1} = 33 \cdot {0.2727272727272727}^{3 - 1} = 33 \cdot {0.2727272727272727}^{2} = 33 \cdot 0.07438016528925619 = 2.454545454545454$

$a_{4} = a_{1} \cdot r^{n - 1} = 33 \cdot {0.2727272727272727}^{4 - 1} = 33 \cdot {0.2727272727272727}^{3} = 33 \cdot 0.020285499624342593 = 0.6694214876033056$

$a_{5} = a_{1} \cdot r^{n - 1} = 33 \cdot {0.2727272727272727}^{5 - 1} = 33 \cdot {0.2727272727272727}^{4} = 33 \cdot 0.005532408988457071 = 0.18256949661908337$

$a_{6} = a_{1} \cdot r^{n - 1} = 33 \cdot {0.2727272727272727}^{6 - 1} = 33 \cdot {0.2727272727272727}^{5} = 33 \cdot 0.0015088388150337464 = 0.04979168089611363$

$a_{7} = a_{1} \cdot r^{n - 1} = 33 \cdot {0.2727272727272727}^{7 - 1} = 33 \cdot {0.2727272727272727}^{6} = 33 \cdot 0.00041150149500920354 = 0.013579549335303717$

$a_{8} = a_{1} \cdot r^{n - 1} = 33 \cdot {0.2727272727272727}^{8 - 1} = 33 \cdot {0.2727272727272727}^{7} = 33 \cdot 0.00011222768045705551 = 0.003703513455082832$

$a_{9} = a_{1} \cdot r^{n - 1} = 33 \cdot {0.2727272727272727}^{9 - 1} = 33 \cdot {0.2727272727272727}^{8} = 33 \cdot 3.060754921556059 E - 05 = 0.0010100491241134995$

$a_{10} = a_{1} \cdot r^{n - 1} = 33 \cdot {0.2727272727272727}^{10 - 1} = 33 \cdot {0.2727272727272727}^{9} = 33 \cdot 8.347513422425615 E - 06 = 0.0002754679429400453$

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几何数列