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解答 - 几何数列

公比是：

r = 3.5185185185185186

r=3.5185185185185186

该系列的和是：

s = 121

s=121

此系列的通用形式是：

a_{n} = 27 \cdot {3.5185185185185186}^{n - 1}

a_n=27*3.5185185185185186^(n-1)

这个序列的第n项是：

27, 95, 334.25925925925924, 1176.0973936899863, 4138.120459279582, 14560.053467835565, 51229.81775719922, 180253.06247903427, 634223.7383521576, 2231527.9682761105

27,95,334.25925925925924,1176.0973936899863,4138.120459279582,14560.053467835565,51229.81775719922,180253.06247903427,634223.7383521576,2231527.9682761105

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：

$a_{2} a_{1} = \frac{95}{27} = 3.5185185185185186$

该序列的公比（ $r$ ）保持不变，并且等于两个连续项的商。
$r = 3.5185185185185186$

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： $a = 27$ 、公比： $r = 3.5185185185185186$ 和元素数目 $n = 2$ 插入几何级数求和公式：

$s_{2} = 27 * ((1 - {3.5185185185185186}^{2}) / (1 - 3.5185185185185186))$

$s_{2} = 27 * ((1 - 12.37997256515775) / (1 - 3.5185185185185186))$

$s_{2} = 27 * (- 11.37997256515775 / (1 - 3.5185185185185186))$

$s_{2} = 27 * (- 11.37997256515775 / - 2.5185185185185186)$

$s_{2} = 27 \cdot 4.518518518518518$

$s_{2} = 121.99999999999999$

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： $a = 27$ 和公比： $r = 3.5185185185185186$ 插入几何级数的公式：

$a_{n} = 27 \cdot {3.5185185185185186}^{n - 1}$

4. 找到第n项

使用通用公式找到第n项

$a_{1} = 27$

$a_{2} = a_{1} \cdot r^{n - 1} = 27 \cdot {3.5185185185185186}^{2 - 1} = 27 \cdot {3.5185185185185186}^{1} = 27 \cdot 3.5185185185185186 = 95$

$a_{3} = a_{1} \cdot r^{n - 1} = 27 \cdot {3.5185185185185186}^{3 - 1} = 27 \cdot {3.5185185185185186}^{2} = 27 \cdot 12.37997256515775 = 334.25925925925924$

$a_{4} = a_{1} \cdot r^{n - 1} = 27 \cdot {3.5185185185185186}^{4 - 1} = 27 \cdot {3.5185185185185186}^{3} = 27 \cdot 43.55916272925875 = 1176.0973936899863$

$a_{5} = a_{1} \cdot r^{n - 1} = 27 \cdot {3.5185185185185186}^{5 - 1} = 27 \cdot {3.5185185185185186}^{4} = 27 \cdot 153.2637207140586 = 4138.120459279582$

$a_{6} = a_{1} \cdot r^{n - 1} = 27 \cdot {3.5185185185185186}^{6 - 1} = 27 \cdot {3.5185185185185186}^{5} = 27 \cdot 539.2612395494654 = 14560.053467835565$

$a_{7} = a_{1} \cdot r^{n - 1} = 27 \cdot {3.5185185185185186}^{7 - 1} = 27 \cdot {3.5185185185185186}^{6} = 27 \cdot 1897.400657674045 = 51229.81775719922$

$a_{8} = a_{1} \cdot r^{n - 1} = 27 \cdot {3.5185185185185186}^{8 - 1} = 27 \cdot {3.5185185185185186}^{7} = 27 \cdot 6676.039351075344 = 180253.06247903427$

$a_{9} = a_{1} \cdot r^{n - 1} = 27 \cdot {3.5185185185185186}^{9 - 1} = 27 \cdot {3.5185185185185186}^{8} = 27 \cdot 23489.76808711695 = 634223.7383521576$

$a_{10} = a_{1} \cdot r^{n - 1} = 27 \cdot {3.5185185185185186}^{10 - 1} = 27 \cdot {3.5185185185185186}^{9} = 27 \cdot 82649.18401022631 = 2231527.9682761105$

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