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解答 - 几何数列

公比是：

r = 0.5181818181818182

r=0.5181818181818182

该系列的和是：

s = 167

s=167

此系列的通用形式是：

a_{n} = 110 \cdot {0.5181818181818182}^{n - 1}

a_n=110*0.5181818181818182^(n-1)

这个序列的第n项是：

110, 57, 29.53636363636364, 15.30520661157025, 7.930879789631857, 4.109637709172871, 2.12953954020776, 1.1034886708349303, 0.5718077657962821, 0.29630038773080075

110,57,29.53636363636364,15.30520661157025,7.930879789631857,4.109637709172871,2.12953954020776,1.1034886708349303,0.5718077657962821,0.29630038773080075

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：

$a_{2} a_{1} = \frac{57}{110} = 0.5181818181818182$

该序列的公比（ $r$ ）保持不变，并且等于两个连续项的商。
$r = 0.5181818181818182$

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： $r = 0.5181818181818182$ 和元素数目 $n = 2$ 插入几何级数求和公式：

$s_{2} = 110 * ((1 - {0.5181818181818182}^{2}) / (1 - 0.5181818181818182))$

$s_{2} = 110 * ((1 - 0.2685123966942149) / (1 - 0.5181818181818182))$

$s_{2} = 110 * (0.7314876033057851 / (1 - 0.5181818181818182))$

$s_{2} = 110 * (0.7314876033057851 / 0.4818181818181818)$

$s_{2} = 110 \cdot 1.518181818181818$

$s_{2} = 167$

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： $r = 0.5181818181818182$ 插入几何级数的公式：

$a_{n} = 110 \cdot {0.5181818181818182}^{n - 1}$

4. 找到第n项

使用通用公式找到第n项

$a_{1} = 110$

$a_{2} = a_{1} \cdot r^{n - 1} = 110 \cdot {0.5181818181818182}^{2 - 1} = 110 \cdot {0.5181818181818182}^{1} = 110 \cdot 0.5181818181818182 = 57$

$a_{3} = a_{1} \cdot r^{n - 1} = 110 \cdot {0.5181818181818182}^{3 - 1} = 110 \cdot {0.5181818181818182}^{2} = 110 \cdot 0.2685123966942149 = 29.53636363636364$

$a_{4} = a_{1} \cdot r^{n - 1} = 110 \cdot {0.5181818181818182}^{4 - 1} = 110 \cdot {0.5181818181818182}^{3} = 110 \cdot 0.1391382419233659 = 15.30520661157025$

$a_{5} = a_{1} \cdot r^{n - 1} = 110 \cdot {0.5181818181818182}^{5 - 1} = 110 \cdot {0.5181818181818182}^{4} = 110 \cdot 0.07209890717847142 = 7.930879789631857$

$a_{6} = a_{1} \cdot r^{n - 1} = 110 \cdot {0.5181818181818182}^{6 - 1} = 110 \cdot {0.5181818181818182}^{5} = 110 \cdot 0.037360342810662465 = 4.109637709172871$

$a_{7} = a_{1} \cdot r^{n - 1} = 110 \cdot {0.5181818181818182}^{7 - 1} = 110 \cdot {0.5181818181818182}^{6} = 110 \cdot 0.019359450365525093 = 2.12953954020776$

$a_{8} = a_{1} \cdot r^{n - 1} = 110 \cdot {0.5181818181818182}^{8 - 1} = 110 \cdot {0.5181818181818182}^{7} = 110 \cdot 0.010031715189408458 = 1.1034886708349303$

$a_{9} = a_{1} \cdot r^{n - 1} = 110 \cdot {0.5181818181818182}^{9 - 1} = 110 \cdot {0.5181818181818182}^{8} = 110 \cdot 0.005198252416329837 = 0.5718077657962821$

$a_{10} = a_{1} \cdot r^{n - 1} = 110 \cdot {0.5181818181818182}^{10 - 1} = 110 \cdot {0.5181818181818182}^{9} = 110 \cdot 0.002693639888461825 = 0.29630038773080075$

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几何数列