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解答 - 几何数列

公比是：

r = 0.2570093457943925

r=0.2570093457943925

该系列的和是：

s = 269

s=269

此系列的通用形式是：

a_{n} = 214 \cdot {0.2570093457943925}^{n - 1}

a_n=214*0.2570093457943925^(n-1)

这个序列的第n项是：

214, 54.99999999999999, 14.135514018691586, 3.6329592104113884, 0.9337044699655437, 0.23997077499114436, 0.061674731890247386, 0.015850982495157033, 0.004073850641278677, 0.0010470176881790991

214,54.99999999999999,14.135514018691586,3.6329592104113884,0.9337044699655437,0.23997077499114436,0.061674731890247386,0.015850982495157033,0.004073850641278677,0.0010470176881790991

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：

$a_{2} a_{1} = \frac{55}{214} = 0.2570093457943925$

该序列的公比（ $r$ ）保持不变，并且等于两个连续项的商。
$r = 0.2570093457943925$

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： $a = 214$ 、公比： $r = 0.2570093457943925$ 和元素数目 $n = 2$ 插入几何级数求和公式：

$s_{2} = 214 * ((1 - {0.2570093457943925}^{2}) / (1 - 0.2570093457943925))$

$s_{2} = 214 * ((1 - 0.06605380382566162) / (1 - 0.2570093457943925))$

$s_{2} = 214 * (0.9339461961743384 / (1 - 0.2570093457943925))$

$s_{2} = 214 * (0.9339461961743384 / 0.7429906542056075)$

$s_{2} = 214 \cdot 1.2570093457943925$

$s_{2} = 269$

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： $a = 214$ 和公比： $r = 0.2570093457943925$ 插入几何级数的公式：

$a_{n} = 214 \cdot {0.2570093457943925}^{n - 1}$

4. 找到第n项

使用通用公式找到第n项

$a_{1} = 214$

$a_{2} = a_{1} \cdot r^{n - 1} = 214 \cdot {0.2570093457943925}^{2 - 1} = 214 \cdot {0.2570093457943925}^{1} = 214 \cdot 0.2570093457943925 = 54.99999999999999$

$a_{3} = a_{1} \cdot r^{n - 1} = 214 \cdot {0.2570093457943925}^{3 - 1} = 214 \cdot {0.2570093457943925}^{2} = 214 \cdot 0.06605380382566162 = 14.135514018691586$

$a_{4} = a_{1} \cdot r^{n - 1} = 214 \cdot {0.2570093457943925}^{4 - 1} = 214 \cdot {0.2570093457943925}^{3} = 214 \cdot 0.016976444908464432 = 3.6329592104113884$

$a_{5} = a_{1} \cdot r^{n - 1} = 214 \cdot {0.2570093457943925}^{5 - 1} = 214 \cdot {0.2570093457943925}^{4} = 214 \cdot 0.004363104999838989 = 0.9337044699655437$

$a_{6} = a_{1} \cdot r^{n - 1} = 214 \cdot {0.2570093457943925}^{6 - 1} = 214 \cdot {0.2570093457943925}^{5} = 214 \cdot 0.0011213587616408615 = 0.23997077499114436$

$a_{7} = a_{1} \cdot r^{n - 1} = 214 \cdot {0.2570093457943925}^{7 - 1} = 214 \cdot {0.2570093457943925}^{6} = 214 \cdot 0.00028819968173012797 = 0.061674731890247386$

$a_{8} = a_{1} \cdot r^{n - 1} = 214 \cdot {0.2570093457943925}^{8 - 1} = 214 \cdot {0.2570093457943925}^{7} = 214 \cdot 7.407001165961231 E - 05 = 0.015850982495157033$

$a_{9} = a_{1} \cdot r^{n - 1} = 214 \cdot {0.2570093457943925}^{9 - 1} = 214 \cdot {0.2570093457943925}^{8} = 214 \cdot 1.9036685239619985 E - 05 = 0.004073850641278677$

$a_{10} = a_{1} \cdot r^{n - 1} = 214 \cdot {0.2570093457943925}^{10 - 1} = 214 \cdot {0.2570093457943925}^{9} = 214 \cdot 4.892606019528501 E - 06 = 0.0010470176881790991$

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几何数列