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解答 - 几何数列

公比是：

r = 2.8518518518518516

r=2.8518518518518516

该系列的和是：

s = 312

s=312

此系列的通用形式是：

a_{n} = 81 \cdot {2.8518518518518516}^{n - 1}

a_n=81*2.8518518518518516^(n-1)

这个序列的第n项是：

81, 230.99999999999997, 658.7777777777777, 1878.736625514403, 5357.878524615149, 15279.875792420977, 43575.94207468205, 124272.13110187101, 354405.7072164469, 1010712.5724320894

81,230.99999999999997,658.7777777777777,1878.736625514403,5357.878524615149,15279.875792420977,43575.94207468205,124272.13110187101,354405.7072164469,1010712.5724320894

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：

$a_{2} a_{1} = \frac{231}{81} = 2.8518518518518516$

该序列的公比（ $r$ ）保持不变，并且等于两个连续项的商。
$r = 2.8518518518518516$

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： $a = 81$ 、公比： $r = 2.8518518518518516$ 和元素数目 $n = 2$ 插入几何级数求和公式：

$s_{2} = 81 * ((1 - {2.8518518518518516}^{2}) / (1 - 2.8518518518518516))$

$s_{2} = 81 * ((1 - 8.133058984910836) / (1 - 2.8518518518518516))$

$s_{2} = 81 * (- 7.133058984910836 / (1 - 2.8518518518518516))$

$s_{2} = 81 * (- 7.133058984910836 / - 1.8518518518518516)$

$s_{2} = 81 \cdot 3.8518518518518516$

$s_{2} = 312$

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： $a = 81$ 和公比： $r = 2.8518518518518516$ 插入几何级数的公式：

$a_{n} = 81 \cdot {2.8518518518518516}^{n - 1}$

4. 找到第n项

使用通用公式找到第n项

$a_{1} = 81$

$a_{2} = a_{1} \cdot r^{n - 1} = 81 \cdot {2.8518518518518516}^{2 - 1} = 81 \cdot {2.8518518518518516}^{1} = 81 \cdot 2.8518518518518516 = 230.99999999999997$

$a_{3} = a_{1} \cdot r^{n - 1} = 81 \cdot {2.8518518518518516}^{3 - 1} = 81 \cdot {2.8518518518518516}^{2} = 81 \cdot 8.133058984910836 = 658.7777777777777$

$a_{4} = a_{1} \cdot r^{n - 1} = 81 \cdot {2.8518518518518516}^{4 - 1} = 81 \cdot {2.8518518518518516}^{3} = 81 \cdot 23.194279327338307 = 1878.736625514403$

$a_{5} = a_{1} \cdot r^{n - 1} = 81 \cdot {2.8518518518518516}^{5 - 1} = 81 \cdot {2.8518518518518516}^{4} = 81 \cdot 66.14664845203887 = 5357.878524615149$

$a_{6} = a_{1} \cdot r^{n - 1} = 81 \cdot {2.8518518518518516}^{6 - 1} = 81 \cdot {2.8518518518518516}^{5} = 81 \cdot 188.64044188174046 = 15279.875792420977$

$a_{7} = a_{1} \cdot r^{n - 1} = 81 \cdot {2.8518518518518516}^{7 - 1} = 81 \cdot {2.8518518518518516}^{6} = 81 \cdot 537.9745935145932 = 43575.94207468205$

$a_{8} = a_{1} \cdot r^{n - 1} = 81 \cdot {2.8518518518518516}^{8 - 1} = 81 \cdot {2.8518518518518516}^{7} = 81 \cdot 1534.2238407638397 = 124272.13110187101$

$a_{9} = a_{1} \cdot r^{n - 1} = 81 \cdot {2.8518518518518516}^{9 - 1} = 81 \cdot {2.8518518518518516}^{8} = 81 \cdot 4375.379101437617 = 354405.7072164469$

$a_{10} = a_{1} \cdot r^{n - 1} = 81 \cdot {2.8518518518518516}^{10 - 1} = 81 \cdot {2.8518518518518516}^{9} = 81 \cdot 12477.932992988757 = 1010712.5724320894$

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