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解答 - 几何数列

公比是：

r = 0.39344262295081966

r=0.39344262295081966

该系列的和是：

s = 85

s=85

此系列的通用形式是：

a_{n} = 61 \cdot {0.39344262295081966}^{n - 1}

a_n=61*0.39344262295081966^(n-1)

这个序列的第n项是：

61, 24, 9.442622950819672, 3.7151303413061005, 1.461690626087646, 0.5750913938705493, 0.2262654664408718, 0.08902247859968727, 0.03502523748184417, 0.013780421304332133

61,24,9.442622950819672,3.7151303413061005,1.461690626087646,0.5750913938705493,0.2262654664408718,0.08902247859968727,0.03502523748184417,0.013780421304332133

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：

$a_{2} a_{1} = \frac{24}{61} = 0.39344262295081966$

该序列的公比（ $r$ ）保持不变，并且等于两个连续项的商。
$r = 0.39344262295081966$

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： $a = 61$ 、公比： $r = 0.39344262295081966$ 和元素数目 $n = 2$ 插入几何级数求和公式：

$s_{2} = 61 * ((1 - {0.39344262295081966}^{2}) / (1 - 0.39344262295081966))$

$s_{2} = 61 * ((1 - 0.15479709755442084) / (1 - 0.39344262295081966))$

$s_{2} = 61 * (0.8452029024455792 / (1 - 0.39344262295081966))$

$s_{2} = 61 * (0.8452029024455792 / 0.6065573770491803)$

$s_{2} = 61 \cdot 1.3934426229508197$

$s_{2} = 85$

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： $a = 61$ 和公比： $r = 0.39344262295081966$ 插入几何级数的公式：

$a_{n} = 61 \cdot {0.39344262295081966}^{n - 1}$

4. 找到第n项

使用通用公式找到第n项

$a_{1} = 61$

$a_{2} = a_{1} \cdot r^{n - 1} = 61 \cdot {0.39344262295081966}^{2 - 1} = 61 \cdot {0.39344262295081966}^{1} = 61 \cdot 0.39344262295081966 = 24$

$a_{3} = a_{1} \cdot r^{n - 1} = 61 \cdot {0.39344262295081966}^{3 - 1} = 61 \cdot {0.39344262295081966}^{2} = 61 \cdot 0.15479709755442084 = 9.442622950819672$

$a_{4} = a_{1} \cdot r^{n - 1} = 61 \cdot {0.39344262295081966}^{4 - 1} = 61 \cdot {0.39344262295081966}^{3} = 61 \cdot 0.06090377608698525 = 3.7151303413061005$

$a_{5} = a_{1} \cdot r^{n - 1} = 61 \cdot {0.39344262295081966}^{5 - 1} = 61 \cdot {0.39344262295081966}^{4} = 61 \cdot 0.023962141411272887 = 1.461690626087646$

$a_{6} = a_{1} \cdot r^{n - 1} = 61 \cdot {0.39344262295081966}^{6 - 1} = 61 \cdot {0.39344262295081966}^{5} = 61 \cdot 0.00942772776836966 = 0.5750913938705493$

$a_{7} = a_{1} \cdot r^{n - 1} = 61 \cdot {0.39344262295081966}^{7 - 1} = 61 \cdot {0.39344262295081966}^{6} = 61 \cdot 0.0037092699416536364 = 0.2262654664408718$

$a_{8} = a_{1} \cdot r^{n - 1} = 61 \cdot {0.39344262295081966}^{8 - 1} = 61 \cdot {0.39344262295081966}^{7} = 61 \cdot 0.0014593848950768405 = 0.08902247859968727$

$a_{9} = a_{1} \cdot r^{n - 1} = 61 \cdot {0.39344262295081966}^{9 - 1} = 61 \cdot {0.39344262295081966}^{8} = 61 \cdot 0.0005741842210138389 = 0.03502523748184417$

$a_{10} = a_{1} \cdot r^{n - 1} = 61 \cdot {0.39344262295081966}^{10 - 1} = 61 \cdot {0.39344262295081966}^{9} = 61 \cdot 0.00022590854597265793 = 0.013780421304332133$

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几何数列