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解答 - 几何数列

公比是：

r = 0.9322033898305084

r=0.9322033898305084

该系列的和是：

s = 113

s=113

此系列的通用形式是：

a_{n} = 59 \cdot {0.9322033898305084}^{n - 1}

a_n=59*0.9322033898305084^(n-1)

这个序列的第n项是：

59, 55, 51.271186440677965, 47.795173800632, 44.55482303448745, 41.53415706604763, 38.718282010722355, 36.09331373880897, 33.64630941753379, 31.36520369431116

59,55,51.271186440677965,47.795173800632,44.55482303448745,41.53415706604763,38.718282010722355,36.09331373880897,33.64630941753379,31.36520369431116

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：

$a_{2} a_{1} = \frac{55}{59} = 0.9322033898305084$

该序列的公比（ $r$ ）保持不变，并且等于两个连续项的商。
$r = 0.9322033898305084$

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： $a = 59$ 、公比： $r = 0.9322033898305084$ 和元素数目 $n = 2$ 插入几何级数求和公式：

$s_{2} = 59 * ((1 - {0.9322033898305084}^{2}) / (1 - 0.9322033898305084))$

$s_{2} = 59 * ((1 - 0.8690031600114909) / (1 - 0.9322033898305084))$

$s_{2} = 59 * (0.13099683998850908 / (1 - 0.9322033898305084))$

$s_{2} = 59 * (0.13099683998850908 / 0.06779661016949157)$

$s_{2} = 59 \cdot 1.9322033898305078$

$s_{2} = 113.99999999999996$

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： $a = 59$ 和公比： $r = 0.9322033898305084$ 插入几何级数的公式：

$a_{n} = 59 \cdot {0.9322033898305084}^{n - 1}$

4. 找到第n项

使用通用公式找到第n项

$a_{1} = 59$

$a_{2} = a_{1} \cdot r^{n - 1} = 59 \cdot {0.9322033898305084}^{2 - 1} = 59 \cdot {0.9322033898305084}^{1} = 59 \cdot 0.9322033898305084 = 55$

$a_{3} = a_{1} \cdot r^{n - 1} = 59 \cdot {0.9322033898305084}^{3 - 1} = 59 \cdot {0.9322033898305084}^{2} = 59 \cdot 0.8690031600114909 = 51.271186440677965$

$a_{4} = a_{1} \cdot r^{n - 1} = 59 \cdot {0.9322033898305084}^{4 - 1} = 59 \cdot {0.9322033898305084}^{3} = 59 \cdot 0.8100876915361356 = 47.795173800632$

$a_{5} = a_{1} \cdot r^{n - 1} = 59 \cdot {0.9322033898305084}^{5 - 1} = 59 \cdot {0.9322033898305084}^{4} = 59 \cdot 0.7551664921099568 = 44.55482303448745$

$a_{6} = a_{1} \cdot r^{n - 1} = 59 \cdot {0.9322033898305084}^{6 - 1} = 59 \cdot {0.9322033898305084}^{5} = 59 \cdot 0.7039687638313157 = 41.53415706604763$

$a_{7} = a_{1} \cdot r^{n - 1} = 59 \cdot {0.9322033898305084}^{7 - 1} = 59 \cdot {0.9322033898305084}^{6} = 59 \cdot 0.6562420679783451 = 38.718282010722355$

$a_{8} = a_{1} \cdot r^{n - 1} = 59 \cdot {0.9322033898305084}^{8 - 1} = 59 \cdot {0.9322033898305084}^{7} = 59 \cdot 0.6117510803187962 = 36.09331373880897$

$a_{9} = a_{1} \cdot r^{n - 1} = 59 \cdot {0.9322033898305084}^{9 - 1} = 59 \cdot {0.9322033898305084}^{8} = 59 \cdot 0.5702764308056575 = 33.64630941753379$

$a_{10} = a_{1} \cdot r^{n - 1} = 59 \cdot {0.9322033898305084}^{10 - 1} = 59 \cdot {0.9322033898305084}^{9} = 59 \cdot 0.5316136219374773 = 31.36520369431116$

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