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解答 - 几何数列

公比是：

r = 0.19047619047619047

r=0.19047619047619047

该系列的和是：

s = 50

s=50

此系列的通用形式是：

a_{n} = 42 \cdot {0.19047619047619047}^{n - 1}

a_n=42*0.19047619047619047^(n-1)

这个序列的第n项是：

42, 8, 1.5238095238095235, 0.2902494331065759, 0.05528560630601446, 0.010530591677336087, 0.0020058269861592546, 0.0003820622830779532, 7.277376820532442 E - 05, 1.3861670134347507 E - 05

42,8,1.5238095238095235,0.2902494331065759,0.05528560630601446,0.010530591677336087,0.0020058269861592546,0.0003820622830779532,7.277376820532442E-05,1.3861670134347507E-05

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：

$a_{2} a_{1} = \frac{8}{42} = 0.19047619047619047$

该序列的公比（ $r$ ）保持不变，并且等于两个连续项的商。
$r = 0.19047619047619047$

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： $a = 42$ 、公比： $r = 0.19047619047619047$ 和元素数目 $n = 2$ 插入几何级数求和公式：

$s_{2} = 42 * ((1 - {0.19047619047619047}^{2}) / (1 - 0.19047619047619047))$

$s_{2} = 42 * ((1 - 0.03628117913832199) / (1 - 0.19047619047619047))$

$s_{2} = 42 * (0.963718820861678 / (1 - 0.19047619047619047))$

$s_{2} = 42 * (0.963718820861678 / 0.8095238095238095)$

$s_{2} = 42 \cdot 1.1904761904761905$

$s_{2} = 50$

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： $a = 42$ 和公比： $r = 0.19047619047619047$ 插入几何级数的公式：

$a_{n} = 42 \cdot {0.19047619047619047}^{n - 1}$

4. 找到第n项

使用通用公式找到第n项

$a_{1} = 42$

$a_{2} = a_{1} \cdot r^{n - 1} = 42 \cdot {0.19047619047619047}^{2 - 1} = 42 \cdot {0.19047619047619047}^{1} = 42 \cdot 0.19047619047619047 = 8$

$a_{3} = a_{1} \cdot r^{n - 1} = 42 \cdot {0.19047619047619047}^{3 - 1} = 42 \cdot {0.19047619047619047}^{2} = 42 \cdot 0.03628117913832199 = 1.5238095238095235$

$a_{4} = a_{1} \cdot r^{n - 1} = 42 \cdot {0.19047619047619047}^{4 - 1} = 42 \cdot {0.19047619047619047}^{3} = 42 \cdot 0.006910700788251807 = 0.2902494331065759$

$a_{5} = a_{1} \cdot r^{n - 1} = 42 \cdot {0.19047619047619047}^{5 - 1} = 42 \cdot {0.19047619047619047}^{4} = 42 \cdot 0.0013163239596670109 = 0.05528560630601446$

$a_{6} = a_{1} \cdot r^{n - 1} = 42 \cdot {0.19047619047619047}^{6 - 1} = 42 \cdot {0.19047619047619047}^{5} = 42 \cdot 0.0002507283732699068 = 0.010530591677336087$

$a_{7} = a_{1} \cdot r^{n - 1} = 42 \cdot {0.19047619047619047}^{7 - 1} = 42 \cdot {0.19047619047619047}^{6} = 42 \cdot 4.7757785384744155 E - 05 = 0.0020058269861592546$

$a_{8} = a_{1} \cdot r^{n - 1} = 42 \cdot {0.19047619047619047}^{8 - 1} = 42 \cdot {0.19047619047619047}^{7} = 42 \cdot 9.096721025665552 E - 06 = 0.0003820622830779532$

$a_{9} = a_{1} \cdot r^{n - 1} = 42 \cdot {0.19047619047619047}^{9 - 1} = 42 \cdot {0.19047619047619047}^{8} = 42 \cdot 1.7327087667934384 E - 06 = 7.277376820532442 E - 05$

$a_{10} = a_{1} \cdot r^{n - 1} = 42 \cdot {0.19047619047619047}^{10 - 1} = 42 \cdot {0.19047619047619047}^{9} = 42 \cdot 3.300397651035121 E - 07 = 1.3861670134347507 E - 05$

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