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解答 - 几何数列

公比是：

r = 0.7272727272727273

r=0.7272727272727273

该系列的和是：

s = 133

s=133

此系列的通用形式是：

a_{n} = 77 \cdot {0.7272727272727273}^{n - 1}

a_n=77*0.7272727272727273^(n-1)

这个序列的第n项是：

77, 56, 40.72727272727273, 29.619834710743802, 21.54169797145004, 15.666689433781848, 11.393955951841344, 8.286513419520977, 6.026555214197074, 4.382949246688782

77,56,40.72727272727273,29.619834710743802,21.54169797145004,15.666689433781848,11.393955951841344,8.286513419520977,6.026555214197074,4.382949246688782

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：

$a_{2} a_{1} = \frac{56}{77} = 0.7272727272727273$

该序列的公比（ $r$ ）保持不变，并且等于两个连续项的商。
$r = 0.7272727272727273$

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： $a = 77$ 、公比： $r = 0.7272727272727273$ 和元素数目 $n = 2$ 插入几何级数求和公式：

$s_{2} = 77 * ((1 - {0.7272727272727273}^{2}) / (1 - 0.7272727272727273))$

$s_{2} = 77 * ((1 - 0.5289256198347108) / (1 - 0.7272727272727273))$

$s_{2} = 77 * (0.47107438016528924 / (1 - 0.7272727272727273))$

$s_{2} = 77 * (0.47107438016528924 / 0.2727272727272727)$

$s_{2} = 77 \cdot 1.7272727272727273$

$s_{2} = 133$

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： $a = 77$ 和公比： $r = 0.7272727272727273$ 插入几何级数的公式：

$a_{n} = 77 \cdot {0.7272727272727273}^{n - 1}$

4. 找到第n项

使用通用公式找到第n项

$a_{1} = 77$

$a_{2} = a_{1} \cdot r^{n - 1} = 77 \cdot {0.7272727272727273}^{2 - 1} = 77 \cdot {0.7272727272727273}^{1} = 77 \cdot 0.7272727272727273 = 56$

$a_{3} = a_{1} \cdot r^{n - 1} = 77 \cdot {0.7272727272727273}^{3 - 1} = 77 \cdot {0.7272727272727273}^{2} = 77 \cdot 0.5289256198347108 = 40.72727272727273$

$a_{4} = a_{1} \cdot r^{n - 1} = 77 \cdot {0.7272727272727273}^{4 - 1} = 77 \cdot {0.7272727272727273}^{3} = 77 \cdot 0.38467317806160783 = 29.619834710743802$

$a_{5} = a_{1} \cdot r^{n - 1} = 77 \cdot {0.7272727272727273}^{5 - 1} = 77 \cdot {0.7272727272727273}^{4} = 77 \cdot 0.279762311317533 = 21.54169797145004$

$a_{6} = a_{1} \cdot r^{n - 1} = 77 \cdot {0.7272727272727273}^{6 - 1} = 77 \cdot {0.7272727272727273}^{5} = 77 \cdot 0.20346349914002398 = 15.666689433781848$

$a_{7} = a_{1} \cdot r^{n - 1} = 77 \cdot {0.7272727272727273}^{7 - 1} = 77 \cdot {0.7272727272727273}^{6} = 77 \cdot 0.14797345392001746 = 11.393955951841344$

$a_{8} = a_{1} \cdot r^{n - 1} = 77 \cdot {0.7272727272727273}^{8 - 1} = 77 \cdot {0.7272727272727273}^{7} = 77 \cdot 0.10761705739637634 = 8.286513419520977$

$a_{9} = a_{1} \cdot r^{n - 1} = 77 \cdot {0.7272727272727273}^{9 - 1} = 77 \cdot {0.7272727272727273}^{8} = 77 \cdot 0.07826695083372824 = 6.026555214197074$

$a_{10} = a_{1} \cdot r^{n - 1} = 77 \cdot {0.7272727272727273}^{10 - 1} = 77 \cdot {0.7272727272727273}^{9} = 77 \cdot 0.056921418788166 = 4.382949246688782$

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