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解答 - 几何数列

公比是：

r = 0.030303030303030304

r=0.030303030303030304

该系列的和是：

s = 102

s=102

此系列的通用形式是：

a_{n} = 99 \cdot {0.030303030303030304}^{n - 1}

a_n=99*0.030303030303030304^(n-1)

这个序列的第n项是：

99, 3, 0.09090909090909093, 0.0027548209366391185, 8.347942232239753 E - 05, 2.529679464315077 E - 06, 7.665695346409325 E - 08, 2.3229379837604015 E - 09, 7.039206011395156 E - 11, 2.133092730725805 E - 12

99,3,0.09090909090909093,0.0027548209366391185,8.347942232239753E-05,2.529679464315077E-06,7.665695346409325E-08,2.3229379837604015E-09,7.039206011395156E-11,2.133092730725805E-12

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：

$a_{2} a_{1} = \frac{3}{99} = 0.030303030303030304$

该序列的公比（ $r$ ）保持不变，并且等于两个连续项的商。
$r = 0.030303030303030304$

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： $a = 99$ 、公比： $r = 0.030303030303030304$ 和元素数目 $n = 2$ 插入几何级数求和公式：

$s_{2} = 99 * ((1 - {0.030303030303030304}^{2}) / (1 - 0.030303030303030304))$

$s_{2} = 99 * ((1 - 0.0009182736455463729) / (1 - 0.030303030303030304))$

$s_{2} = 99 * (0.9990817263544536 / (1 - 0.030303030303030304))$

$s_{2} = 99 * (0.9990817263544536 / 0.9696969696969697)$

$s_{2} = 99 \cdot 1.0303030303030303$

$s_{2} = 102$

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： $a = 99$ 和公比： $r = 0.030303030303030304$ 插入几何级数的公式：

$a_{n} = 99 \cdot {0.030303030303030304}^{n - 1}$

4. 找到第n项

使用通用公式找到第n项

$a_{1} = 99$

$a_{2} = a_{1} \cdot r^{n - 1} = 99 \cdot {0.030303030303030304}^{2 - 1} = 99 \cdot {0.030303030303030304}^{1} = 99 \cdot 0.030303030303030304 = 3$

$a_{3} = a_{1} \cdot r^{n - 1} = 99 \cdot {0.030303030303030304}^{3 - 1} = 99 \cdot {0.030303030303030304}^{2} = 99 \cdot 0.0009182736455463729 = 0.09090909090909093$

$a_{4} = a_{1} \cdot r^{n - 1} = 99 \cdot {0.030303030303030304}^{4 - 1} = 99 \cdot {0.030303030303030304}^{3} = 99 \cdot 2.7826474107465846 E - 05 = 0.0027548209366391185$

$a_{5} = a_{1} \cdot r^{n - 1} = 99 \cdot {0.030303030303030304}^{5 - 1} = 99 \cdot {0.030303030303030304}^{4} = 99 \cdot 8.432264881050256 E - 07 = 8.347942232239753 E - 05$

$a_{6} = a_{1} \cdot r^{n - 1} = 99 \cdot {0.030303030303030304}^{6 - 1} = 99 \cdot {0.030303030303030304}^{5} = 99 \cdot 2.5552317821364412 E - 08 = 2.529679464315077 E - 06$

$a_{7} = a_{1} \cdot r^{n - 1} = 99 \cdot {0.030303030303030304}^{7 - 1} = 99 \cdot {0.030303030303030304}^{6} = 99 \cdot 7.743126612534672 E - 10 = 7.665695346409325 E - 08$

$a_{8} = a_{1} \cdot r^{n - 1} = 99 \cdot {0.030303030303030304}^{8 - 1} = 99 \cdot {0.030303030303030304}^{7} = 99 \cdot 2.3464020037983852 E - 11 = 2.3229379837604015 E - 09$

$a_{9} = a_{1} \cdot r^{n - 1} = 99 \cdot {0.030303030303030304}^{9 - 1} = 99 \cdot {0.030303030303030304}^{8} = 99 \cdot 7.110309102419349 E - 13 = 7.039206011395156 E - 11$

$a_{10} = a_{1} \cdot r^{n - 1} = 99 \cdot {0.030303030303030304}^{10 - 1} = 99 \cdot {0.030303030303030304}^{9} = 99 \cdot 2.1546391219452575 E - 14 = 2.133092730725805 E - 12$

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几何数列