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解答 - 几何数列

公比是：

r = 0.3384615384615385

r=0.3384615384615385

该系列的和是：

s = 87

s=87

此系列的通用形式是：

a_{n} = 65 \cdot {0.3384615384615385}^{n - 1}

a_n=65*0.3384615384615385^(n-1)

这个序列的第n项是：

65, 22, 7.446153846153847, 2.520236686390533, 0.8530031861629497, 0.288708770701306, 0.09771681469890359, 0.033073383436551994, 0.01119406824006375, 0.0037887615581754233

65,22,7.446153846153847,2.520236686390533,0.8530031861629497,0.288708770701306,0.09771681469890359,0.033073383436551994,0.01119406824006375,0.0037887615581754233

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：

$a_{2} a_{1} = \frac{22}{65} = 0.3384615384615385$

该序列的公比（ $r$ ）保持不变，并且等于两个连续项的商。
$r = 0.3384615384615385$

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： $r = 0.3384615384615385$ 和元素数目 $n = 2$ 插入几何级数求和公式：

$s_{2} = 65 * ((1 - {0.3384615384615385}^{2}) / (1 - 0.3384615384615385))$

$s_{2} = 65 * ((1 - 0.1145562130177515) / (1 - 0.3384615384615385))$

$s_{2} = 65 * (0.8854437869822485 / (1 - 0.3384615384615385))$

$s_{2} = 65 * (0.8854437869822485 / 0.6615384615384615)$

$s_{2} = 65 \cdot 1.3384615384615384$

$s_{2} = 87$

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： $r = 0.3384615384615385$ 插入几何级数的公式：

$a_{n} = 65 \cdot {0.3384615384615385}^{n - 1}$

4. 找到第n项

使用通用公式找到第n项

$a_{1} = 65$

$a_{2} = a_{1} \cdot r^{n - 1} = 65 \cdot {0.3384615384615385}^{2 - 1} = 65 \cdot {0.3384615384615385}^{1} = 65 \cdot 0.3384615384615385 = 22$

$a_{3} = a_{1} \cdot r^{n - 1} = 65 \cdot {0.3384615384615385}^{3 - 1} = 65 \cdot {0.3384615384615385}^{2} = 65 \cdot 0.1145562130177515 = 7.446153846153847$

$a_{4} = a_{1} \cdot r^{n - 1} = 65 \cdot {0.3384615384615385}^{4 - 1} = 65 \cdot {0.3384615384615385}^{3} = 65 \cdot 0.03877287209831589 = 2.520236686390533$

$a_{5} = a_{1} \cdot r^{n - 1} = 65 \cdot {0.3384615384615385}^{5 - 1} = 65 \cdot {0.3384615384615385}^{4} = 65 \cdot 0.013123125940968457 = 0.8530031861629497$

$a_{6} = a_{1} \cdot r^{n - 1} = 65 \cdot {0.3384615384615385}^{6 - 1} = 65 \cdot {0.3384615384615385}^{5} = 65 \cdot 0.004441673395404708 = 0.288708770701306$

$a_{7} = a_{1} \cdot r^{n - 1} = 65 \cdot {0.3384615384615385}^{7 - 1} = 65 \cdot {0.3384615384615385}^{6} = 65 \cdot 0.001503335610752363 = 0.09771681469890359$

$a_{8} = a_{1} \cdot r^{n - 1} = 65 \cdot {0.3384615384615385}^{8 - 1} = 65 \cdot {0.3384615384615385}^{7} = 65 \cdot 0.0005088212836392614 = 0.033073383436551994$

$a_{9} = a_{1} \cdot r^{n - 1} = 65 \cdot {0.3384615384615385}^{9 - 1} = 65 \cdot {0.3384615384615385}^{8} = 65 \cdot 0.00017221643446251923 = 0.01119406824006375$

$a_{10} = a_{1} \cdot r^{n - 1} = 65 \cdot {0.3384615384615385}^{10 - 1} = 65 \cdot {0.3384615384615385}^{9} = 65 \cdot 5.8288639356544976 E - 05 = 0.0037887615581754233$

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几何数列