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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 = 150

s=150

此系列的通用形式是：

a_{n} = 110 \cdot {0.36363636363636365}^{n - 1}

a_n=110*0.36363636363636365^(n-1)

这个序列的第n项是：

110, 40, 14.545454545454547, 5.289256198347108, 1.9233658903080393, 0.6994057782938324, 0.25432937392503, 0.09248340870001091, 0.0336303304363676, 0.012229211067770038

110,40,14.545454545454547,5.289256198347108,1.9233658903080393,0.6994057782938324,0.25432937392503,0.09248340870001091,0.0336303304363676,0.012229211067770038

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：

$a_{2} a_{1} = \frac{40}{110} = 0.36363636363636365$

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： $a = 110$ 、公比： $r = 0.36363636363636365$ 和元素数目 $n = 2$ 插入几何级数求和公式：

$s_{2} = 110 * ((1 - {0.36363636363636365}^{2}) / (1 - 0.36363636363636365))$

$s_{2} = 110 * ((1 - 0.1322314049586777) / (1 - 0.36363636363636365))$

$s_{2} = 110 * (0.8677685950413223 / (1 - 0.36363636363636365))$

$s_{2} = 110 * (0.8677685950413223 / 0.6363636363636364)$

$s_{2} = 110 \cdot 1.3636363636363638$

$s_{2} = 150$

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： $a = 110$ 和公比： $r = 0.36363636363636365$ 插入几何级数的公式：

$a_{n} = 110 \cdot {0.36363636363636365}^{n - 1}$

4. 找到第n项

使用通用公式找到第n项

$a_{1} = 110$

$a_{2} = a_{1} \cdot r^{n - 1} = 110 \cdot {0.36363636363636365}^{2 - 1} = 110 \cdot {0.36363636363636365}^{1} = 110 \cdot 0.36363636363636365 = 40$

$a_{3} = a_{1} \cdot r^{n - 1} = 110 \cdot {0.36363636363636365}^{3 - 1} = 110 \cdot {0.36363636363636365}^{2} = 110 \cdot 0.1322314049586777 = 14.545454545454547$

$a_{4} = a_{1} \cdot r^{n - 1} = 110 \cdot {0.36363636363636365}^{4 - 1} = 110 \cdot {0.36363636363636365}^{3} = 110 \cdot 0.04808414725770098 = 5.289256198347108$

$a_{5} = a_{1} \cdot r^{n - 1} = 110 \cdot {0.36363636363636365}^{5 - 1} = 110 \cdot {0.36363636363636365}^{4} = 110 \cdot 0.01748514445734581 = 1.9233658903080393$

$a_{6} = a_{1} \cdot r^{n - 1} = 110 \cdot {0.36363636363636365}^{6 - 1} = 110 \cdot {0.36363636363636365}^{5} = 110 \cdot 0.0063582343481257495 = 0.6994057782938324$

$a_{7} = a_{1} \cdot r^{n - 1} = 110 \cdot {0.36363636363636365}^{7 - 1} = 110 \cdot {0.36363636363636365}^{6} = 110 \cdot 0.0023120852175002727 = 0.25432937392503$

$a_{8} = a_{1} \cdot r^{n - 1} = 110 \cdot {0.36363636363636365}^{8 - 1} = 110 \cdot {0.36363636363636365}^{7} = 110 \cdot 0.0008407582609091901 = 0.09248340870001091$

$a_{9} = a_{1} \cdot r^{n - 1} = 110 \cdot {0.36363636363636365}^{9 - 1} = 110 \cdot {0.36363636363636365}^{8} = 110 \cdot 0.00030573027669425094 = 0.0336303304363676$

$a_{10} = a_{1} \cdot r^{n - 1} = 110 \cdot {0.36363636363636365}^{10 - 1} = 110 \cdot {0.36363636363636365}^{9} = 110 \cdot 0.00011117464607063672 = 0.012229211067770038$

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