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解答 - 几何数列

公比是：

r = 5.493827160493828

r=5.493827160493828

该系列的和是：

s = 526

s=526

此系列的通用形式是：

a_{n} = 81 \cdot {5.493827160493828}^{n - 1}

a_n=81*5.493827160493828^(n-1)

这个序列的第n项是：

81, 445.00000000000006, 2444.7530864197533, 13431.050906873956, 73787.8722661594, 405377.8167708757, 2227075.660037527, 12235168.749588884, 67217902.38971671, 369283537.8200487

81,445.00000000000006,2444.7530864197533,13431.050906873956,73787.8722661594,405377.8167708757,2227075.660037527,12235168.749588884,67217902.38971671,369283537.8200487

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：

$a_{2} a_{1} = \frac{445}{81} = 5.493827160493828$

该序列的公比（ $r$ ）保持不变，并且等于两个连续项的商。
$r = 5.493827160493828$

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： $r = 5.493827160493828$ 和元素数目 $n = 2$ 插入几何级数求和公式：

$s_{2} = 81 * ((1 - {5.493827160493828}^{2}) / (1 - 5.493827160493828))$

$s_{2} = 81 * ((1 - 30.182136869379672) / (1 - 5.493827160493828))$

$s_{2} = 81 * (- 29.182136869379672 / (1 - 5.493827160493828))$

$s_{2} = 81 * (- 29.182136869379672 / - 4.493827160493828)$

$s_{2} = 81 \cdot 6.493827160493828$

$s_{2} = 526$

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： $r = 5.493827160493828$ 插入几何级数的公式：

$a_{n} = 81 \cdot {5.493827160493828}^{n - 1}$

4. 找到第n项

使用通用公式找到第n项

$a_{1} = 81$

$a_{2} = a_{1} \cdot r^{n - 1} = 81 \cdot {5.493827160493828}^{2 - 1} = 81 \cdot {5.493827160493828}^{1} = 81 \cdot 5.493827160493828 = 445.00000000000006$

$a_{3} = a_{1} \cdot r^{n - 1} = 81 \cdot {5.493827160493828}^{3 - 1} = 81 \cdot {5.493827160493828}^{2} = 81 \cdot 30.182136869379672 = 2444.7530864197533$

$a_{4} = a_{1} \cdot r^{n - 1} = 81 \cdot {5.493827160493828}^{4 - 1} = 81 \cdot {5.493827160493828}^{3} = 81 \cdot 165.8154432947402 = 13431.050906873956$

$a_{5} = a_{1} \cdot r^{n - 1} = 81 \cdot {5.493827160493828}^{5 - 1} = 81 \cdot {5.493827160493828}^{4} = 81 \cdot 910.9613860019678 = 73787.8722661594$

$a_{6} = a_{1} \cdot r^{n - 1} = 81 \cdot {5.493827160493828}^{6 - 1} = 81 \cdot {5.493827160493828}^{5} = 81 \cdot 5004.6644045787125 = 405377.8167708757$

$a_{7} = a_{1} \cdot r^{n - 1} = 81 \cdot {5.493827160493828}^{7 - 1} = 81 \cdot {5.493827160493828}^{6} = 81 \cdot 27494.761235031197 = 2227075.660037527$

$a_{8} = a_{1} \cdot r^{n - 1} = 81 \cdot {5.493827160493828}^{8 - 1} = 81 \cdot {5.493827160493828}^{7} = 81 \cdot 151051.46604430722 = 12235168.749588884$

$a_{9} = a_{1} \cdot r^{n - 1} = 81 \cdot {5.493827160493828}^{9 - 1} = 81 \cdot {5.493827160493828}^{8} = 81 \cdot 829850.6467866261 = 67217902.38971671$

$a_{10} = a_{1} \cdot r^{n - 1} = 81 \cdot {5.493827160493828}^{10 - 1} = 81 \cdot {5.493827160493828}^{9} = 81 \cdot 4559056.022469737 = 369283537.8200487$

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