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解答 - 几何数列

公比是：

r = 0.36363636363636365

r=0.36363636363636365

该系列的和是：

s = 180

s=180

此系列的通用形式是：

a_{n} = 132 \cdot {0.36363636363636365}^{n - 1}

a_n=132*0.36363636363636365^(n-1)

这个序列的第n项是：

132, 48, 17.454545454545453, 6.347107438016529, 2.3080390683696472, 0.8392869339525989, 0.305195248710036, 0.1109800904400131, 0.040356396523641126, 0.014675053281324046

132,48,17.454545454545453,6.347107438016529,2.3080390683696472,0.8392869339525989,0.305195248710036,0.1109800904400131,0.040356396523641126,0.014675053281324046

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：

$a_{2} a_{1} = \frac{48}{132} = 0.36363636363636365$

该序列的公比（ $r$ ）保持不变，并且等于两个连续项的商。
$r = 0.36363636363636365$

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： $a = 132$ 、公比： $r = 0.36363636363636365$ 和元素数目 $n = 2$ 插入几何级数求和公式：

$s_{2} = 132 * ((1 - {0.36363636363636365}^{2}) / (1 - 0.36363636363636365))$

$s_{2} = 132 * ((1 - 0.1322314049586777) / (1 - 0.36363636363636365))$

$s_{2} = 132 * (0.8677685950413223 / (1 - 0.36363636363636365))$

$s_{2} = 132 * (0.8677685950413223 / 0.6363636363636364)$

$s_{2} = 132 \cdot 1.3636363636363638$

$s_{2} = 180.00000000000003$

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： $a = 132$ 和公比： $r = 0.36363636363636365$ 插入几何级数的公式：

$a_{n} = 132 \cdot {0.36363636363636365}^{n - 1}$

4. 找到第n项

使用通用公式找到第n项

$a_{1} = 132$

$a_{2} = a_{1} \cdot r^{n - 1} = 132 \cdot {0.36363636363636365}^{2 - 1} = 132 \cdot {0.36363636363636365}^{1} = 132 \cdot 0.36363636363636365 = 48$

$a_{3} = a_{1} \cdot r^{n - 1} = 132 \cdot {0.36363636363636365}^{3 - 1} = 132 \cdot {0.36363636363636365}^{2} = 132 \cdot 0.1322314049586777 = 17.454545454545453$

$a_{4} = a_{1} \cdot r^{n - 1} = 132 \cdot {0.36363636363636365}^{4 - 1} = 132 \cdot {0.36363636363636365}^{3} = 132 \cdot 0.04808414725770098 = 6.347107438016529$

$a_{5} = a_{1} \cdot r^{n - 1} = 132 \cdot {0.36363636363636365}^{5 - 1} = 132 \cdot {0.36363636363636365}^{4} = 132 \cdot 0.01748514445734581 = 2.3080390683696472$

$a_{6} = a_{1} \cdot r^{n - 1} = 132 \cdot {0.36363636363636365}^{6 - 1} = 132 \cdot {0.36363636363636365}^{5} = 132 \cdot 0.0063582343481257495 = 0.8392869339525989$

$a_{7} = a_{1} \cdot r^{n - 1} = 132 \cdot {0.36363636363636365}^{7 - 1} = 132 \cdot {0.36363636363636365}^{6} = 132 \cdot 0.0023120852175002727 = 0.305195248710036$

$a_{8} = a_{1} \cdot r^{n - 1} = 132 \cdot {0.36363636363636365}^{8 - 1} = 132 \cdot {0.36363636363636365}^{7} = 132 \cdot 0.0008407582609091901 = 0.1109800904400131$

$a_{9} = a_{1} \cdot r^{n - 1} = 132 \cdot {0.36363636363636365}^{9 - 1} = 132 \cdot {0.36363636363636365}^{8} = 132 \cdot 0.00030573027669425094 = 0.040356396523641126$

$a_{10} = a_{1} \cdot r^{n - 1} = 132 \cdot {0.36363636363636365}^{10 - 1} = 132 \cdot {0.36363636363636365}^{9} = 132 \cdot 0.00011117464607063672 = 0.014675053281324046$

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几何数列