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解答 - 几何数列

公比是：

r = 0.4153846153846154

r=0.4153846153846154

该系列的和是：

s = 92

s=92

此系列的通用形式是：

a_{n} = 65 \cdot {0.4153846153846154}^{n - 1}

a_n=65*0.4153846153846154^(n-1)

这个序列的第n项是：

65, 27, 11.215384615384616, 4.658698224852072, 1.9351515703231685, 0.8038321907496239, 0.3338995253883053, 0.13869672593052682, 0.0576124861557573, 0.023931340403160726

65,27,11.215384615384616,4.658698224852072,1.9351515703231685,0.8038321907496239,0.3338995253883053,0.13869672593052682,0.0576124861557573,0.023931340403160726

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：

$a_{2} a_{1} = \frac{27}{65} = 0.4153846153846154$

该序列的公比（ $r$ ）保持不变，并且等于两个连续项的商。
$r = 0.4153846153846154$

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： $a = 65$ 、公比： $r = 0.4153846153846154$ 和元素数目 $n = 2$ 插入几何级数求和公式：

$s_{2} = 65 * ((1 - {0.4153846153846154}^{2}) / (1 - 0.4153846153846154))$

$s_{2} = 65 * ((1 - 0.17254437869822487) / (1 - 0.4153846153846154))$

$s_{2} = 65 * (0.8274556213017752 / (1 - 0.4153846153846154))$

$s_{2} = 65 * (0.8274556213017752 / 0.5846153846153845)$

$s_{2} = 65 \cdot 1.4153846153846157$

$s_{2} = 92.00000000000001$

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： $a = 65$ 和公比： $r = 0.4153846153846154$ 插入几何级数的公式：

$a_{n} = 65 \cdot {0.4153846153846154}^{n - 1}$

4. 找到第n项

使用通用公式找到第n项

$a_{1} = 65$

$a_{2} = a_{1} \cdot r^{n - 1} = 65 \cdot {0.4153846153846154}^{2 - 1} = 65 \cdot {0.4153846153846154}^{1} = 65 \cdot 0.4153846153846154 = 27$

$a_{3} = a_{1} \cdot r^{n - 1} = 65 \cdot {0.4153846153846154}^{3 - 1} = 65 \cdot {0.4153846153846154}^{2} = 65 \cdot 0.17254437869822487 = 11.215384615384616$

$a_{4} = a_{1} \cdot r^{n - 1} = 65 \cdot {0.4153846153846154}^{4 - 1} = 65 \cdot {0.4153846153846154}^{3} = 65 \cdot 0.07167228038233957 = 4.658698224852072$

$a_{5} = a_{1} \cdot r^{n - 1} = 65 \cdot {0.4153846153846154}^{5 - 1} = 65 \cdot {0.4153846153846154}^{4} = 65 \cdot 0.029771562620356438 = 1.9351515703231685$

$a_{6} = a_{1} \cdot r^{n - 1} = 65 \cdot {0.4153846153846154}^{6 - 1} = 65 \cdot {0.4153846153846154}^{5} = 65 \cdot 0.012366649088455752 = 0.8038321907496239$

$a_{7} = a_{1} \cdot r^{n - 1} = 65 \cdot {0.4153846153846154}^{7 - 1} = 65 \cdot {0.4153846153846154}^{6} = 65 \cdot 0.005136915775204697 = 0.3338995253883053$

$a_{8} = a_{1} \cdot r^{n - 1} = 65 \cdot {0.4153846153846154}^{8 - 1} = 65 \cdot {0.4153846153846154}^{7} = 65 \cdot 0.0021337957835465666 = 0.13869672593052682$

$a_{9} = a_{1} \cdot r^{n - 1} = 65 \cdot {0.4153846153846154}^{9 - 1} = 65 \cdot {0.4153846153846154}^{8} = 65 \cdot 0.0008863459408578046 = 0.0576124861557573$

$a_{10} = a_{1} \cdot r^{n - 1} = 65 \cdot {0.4153846153846154}^{10 - 1} = 65 \cdot {0.4153846153846154}^{9} = 65 \cdot 0.00036817446774093423 = 0.023931340403160726$

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几何数列