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解答 - 几何数列

公比是：

r = 0.46153846153846156

r=0.46153846153846156

该系列的和是：

s = 475

s=475

此系列的通用形式是：

a_{n} = 325 \cdot {0.46153846153846156}^{n - 1}

a_n=325*0.46153846153846156^(n-1)

这个序列的第n项是：

325, 150, 69.23076923076924, 31.952662721893493, 14.747382794720076, 6.806484366793881, 3.1414543231356378, 1.4499019952933714, 0.6691855362892485, 0.30885486290273007

325,150,69.23076923076924,31.952662721893493,14.747382794720076,6.806484366793881,3.1414543231356378,1.4499019952933714,0.6691855362892485,0.30885486290273007

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：

$a_{2} a_{1} = \frac{150}{325} = 0.46153846153846156$

该序列的公比（ $r$ ）保持不变，并且等于两个连续项的商。
$r = 0.46153846153846156$

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： $a = 325$ 、公比： $r = 0.46153846153846156$ 和元素数目 $n = 2$ 插入几何级数求和公式：

$s_{2} = 325 * ((1 - {0.46153846153846156}^{2}) / (1 - 0.46153846153846156))$

$s_{2} = 325 * ((1 - 0.21301775147928997) / (1 - 0.46153846153846156))$

$s_{2} = 325 * (0.7869822485207101 / (1 - 0.46153846153846156))$

$s_{2} = 325 * (0.7869822485207101 / 0.5384615384615384)$

$s_{2} = 325 \cdot 1.4615384615384617$

$s_{2} = 475.00000000000006$

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： $a = 325$ 和公比： $r = 0.46153846153846156$ 插入几何级数的公式：

$a_{n} = 325 \cdot {0.46153846153846156}^{n - 1}$

4. 找到第n项

使用通用公式找到第n项

$a_{1} = 325$

$a_{2} = a_{1} \cdot r^{n - 1} = 325 \cdot {0.46153846153846156}^{2 - 1} = 325 \cdot {0.46153846153846156}^{1} = 325 \cdot 0.46153846153846156 = 150$

$a_{3} = a_{1} \cdot r^{n - 1} = 325 \cdot {0.46153846153846156}^{3 - 1} = 325 \cdot {0.46153846153846156}^{2} = 325 \cdot 0.21301775147928997 = 69.23076923076924$

$a_{4} = a_{1} \cdot r^{n - 1} = 325 \cdot {0.46153846153846156}^{4 - 1} = 325 \cdot {0.46153846153846156}^{3} = 325 \cdot 0.09831588529813383 = 31.952662721893493$

$a_{5} = a_{1} \cdot r^{n - 1} = 325 \cdot {0.46153846153846156}^{5 - 1} = 325 \cdot {0.46153846153846156}^{4} = 325 \cdot 0.04537656244529254 = 14.747382794720076$

$a_{6} = a_{1} \cdot r^{n - 1} = 325 \cdot {0.46153846153846156}^{6 - 1} = 325 \cdot {0.46153846153846156}^{5} = 325 \cdot 0.02094302882090425 = 6.806484366793881$

$a_{7} = a_{1} \cdot r^{n - 1} = 325 \cdot {0.46153846153846156}^{7 - 1} = 325 \cdot {0.46153846153846156}^{6} = 325 \cdot 0.009666013301955809 = 3.1414543231356378$

$a_{8} = a_{1} \cdot r^{n - 1} = 325 \cdot {0.46153846153846156}^{8 - 1} = 325 \cdot {0.46153846153846156}^{7} = 325 \cdot 0.004461236908594989 = 1.4499019952933714$

$a_{9} = a_{1} \cdot r^{n - 1} = 325 \cdot {0.46153846153846156}^{9 - 1} = 325 \cdot {0.46153846153846156}^{8} = 325 \cdot 0.0020590324193515337 = 0.6691855362892485$

$a_{10} = a_{1} \cdot r^{n - 1} = 325 \cdot {0.46153846153846156}^{10 - 1} = 325 \cdot {0.46153846153846156}^{9} = 325 \cdot 0.0009503226550853232 = 0.30885486290273007$

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几何数列