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解答 - 几何数列

公比是：

r = 0.25925925925925924

r=0.25925925925925924

该系列的和是：

s = 102

s=102

此系列的通用形式是：

a_{n} = 81 \cdot {0.25925925925925924}^{n - 1}

a_n=81*0.25925925925925924^(n-1)

这个序列的第n项是：

81, 21, 5.444444444444444, 1.4115226337448556, 0.36595031245237003, 0.09487600693209591, 0.02459748327869153, 0.006377125294475582, 0.001653328780049225, 0.0004286407948275768

81,21,5.444444444444444,1.4115226337448556,0.36595031245237003,0.09487600693209591,0.02459748327869153,0.006377125294475582,0.001653328780049225,0.0004286407948275768

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：

$a_{2} a_{1} = \frac{21}{81} = 0.25925925925925924$

该序列的公比（ $r$ ）保持不变，并且等于两个连续项的商。
$r = 0.25925925925925924$

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： $r = 0.25925925925925924$ 和元素数目 $n = 2$ 插入几何级数求和公式：

$s_{2} = 81 * ((1 - {0.25925925925925924}^{2}) / (1 - 0.25925925925925924))$

$s_{2} = 81 * ((1 - 0.0672153635116598) / (1 - 0.25925925925925924))$

$s_{2} = 81 * (0.9327846364883402 / (1 - 0.25925925925925924))$

$s_{2} = 81 * (0.9327846364883402 / 0.7407407407407407)$

$s_{2} = 81 \cdot 1.2592592592592593$

$s_{2} = 102$

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： $r = 0.25925925925925924$ 插入几何级数的公式：

$a_{n} = 81 \cdot {0.25925925925925924}^{n - 1}$

4. 找到第n项

使用通用公式找到第n项

$a_{1} = 81$

$a_{2} = a_{1} \cdot r^{n - 1} = 81 \cdot {0.25925925925925924}^{2 - 1} = 81 \cdot {0.25925925925925924}^{1} = 81 \cdot 0.25925925925925924 = 21$

$a_{3} = a_{1} \cdot r^{n - 1} = 81 \cdot {0.25925925925925924}^{3 - 1} = 81 \cdot {0.25925925925925924}^{2} = 81 \cdot 0.0672153635116598 = 5.444444444444444$

$a_{4} = a_{1} \cdot r^{n - 1} = 81 \cdot {0.25925925925925924}^{4 - 1} = 81 \cdot {0.25925925925925924}^{3} = 81 \cdot 0.01742620535487476 = 1.4115226337448556$

$a_{5} = a_{1} \cdot r^{n - 1} = 81 \cdot {0.25925925925925924}^{5 - 1} = 81 \cdot {0.25925925925925924}^{4} = 81 \cdot 0.004517905092004568 = 0.36595031245237003$

$a_{6} = a_{1} \cdot r^{n - 1} = 81 \cdot {0.25925925925925924}^{6 - 1} = 81 \cdot {0.25925925925925924}^{5} = 81 \cdot 0.0011713087275567397 = 0.09487600693209591$

$a_{7} = a_{1} \cdot r^{n - 1} = 81 \cdot {0.25925925925925924}^{7 - 1} = 81 \cdot {0.25925925925925924}^{6} = 81 \cdot 0.0003036726330702658 = 0.02459748327869153$

$a_{8} = a_{1} \cdot r^{n - 1} = 81 \cdot {0.25925925925925924}^{8 - 1} = 81 \cdot {0.25925925925925924}^{7} = 81 \cdot 7.872994190710595 E - 05 = 0.006377125294475582$

$a_{9} = a_{1} \cdot r^{n - 1} = 81 \cdot {0.25925925925925924}^{9 - 1} = 81 \cdot {0.25925925925925924}^{8} = 81 \cdot 2.04114664203608 E - 05 = 0.001653328780049225$

$a_{10} = a_{1} \cdot r^{n - 1} = 81 \cdot {0.25925925925925924}^{10 - 1} = 81 \cdot {0.25925925925925924}^{9} = 81 \cdot 5.291861664537985 E - 06 = 0.0004286407948275768$

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几何数列