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解答 - 几何数列

公比是：

r = 0.5675675675675675

r=0.5675675675675675

该系列的和是：

s = 116

s=116

此系列的通用形式是：

a_{n} = 74 \cdot {0.5675675675675675}^{n - 1}

a_n=74*0.5675675675675675^(n-1)

这个序列的第n项是：

74, 42, 23.837837837837835, 13.529583637691744, 7.678952875446665, 4.358324604983243, 2.4736436947202187, 1.403959934841205, 0.7968421251801433, 0.4522617467238651

74,42,23.837837837837835,13.529583637691744,7.678952875446665,4.358324604983243,2.4736436947202187,1.403959934841205,0.7968421251801433,0.4522617467238651

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：

$a_{2} a_{1} = \frac{42}{74} = 0.5675675675675675$

该序列的公比（ $r$ ）保持不变，并且等于两个连续项的商。
$r = 0.5675675675675675$

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： $a = 74$ 、公比： $r = 0.5675675675675675$ 和元素数目 $n = 2$ 插入几何级数求和公式：

$s_{2} = 74 * ((1 - {0.5675675675675675}^{2}) / (1 - 0.5675675675675675))$

$s_{2} = 74 * ((1 - 0.32213294375456536) / (1 - 0.5675675675675675))$

$s_{2} = 74 * (0.6778670562454346 / (1 - 0.5675675675675675))$

$s_{2} = 74 * (0.6778670562454346 / 0.43243243243243246)$

$s_{2} = 74 \cdot 1.5675675675675675$

$s_{2} = 116$

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： $a = 74$ 和公比： $r = 0.5675675675675675$ 插入几何级数的公式：

$a_{n} = 74 \cdot {0.5675675675675675}^{n - 1}$

4. 找到第n项

使用通用公式找到第n项

$a_{1} = 74$

$a_{2} = a_{1} \cdot r^{n - 1} = 74 \cdot {0.5675675675675675}^{2 - 1} = 74 \cdot {0.5675675675675675}^{1} = 74 \cdot 0.5675675675675675 = 42$

$a_{3} = a_{1} \cdot r^{n - 1} = 74 \cdot {0.5675675675675675}^{3 - 1} = 74 \cdot {0.5675675675675675}^{2} = 74 \cdot 0.32213294375456536 = 23.837837837837835$

$a_{4} = a_{1} \cdot r^{n - 1} = 74 \cdot {0.5675675675675675}^{4 - 1} = 74 \cdot {0.5675675675675675}^{3} = 74 \cdot 0.1828322113201587 = 13.529583637691744$

$a_{5} = a_{1} \cdot r^{n - 1} = 74 \cdot {0.5675675675675675}^{5 - 1} = 74 \cdot {0.5675675675675675}^{4} = 74 \cdot 0.10376963345198197 = 7.678952875446665$

$a_{6} = a_{1} \cdot r^{n - 1} = 74 \cdot {0.5675675675675675}^{6 - 1} = 74 \cdot {0.5675675675675675}^{5} = 74 \cdot 0.05889627844571949 = 4.358324604983243$

$a_{7} = a_{1} \cdot r^{n - 1} = 74 \cdot {0.5675675675675675}^{7 - 1} = 74 \cdot {0.5675675675675675}^{6} = 74 \cdot 0.03342761749621917 = 2.4736436947202187$

$a_{8} = a_{1} \cdot r^{n - 1} = 74 \cdot {0.5675675675675675}^{8 - 1} = 74 \cdot {0.5675675675675675}^{7} = 74 \cdot 0.018972431551908177 = 1.403959934841205$

$a_{9} = a_{1} \cdot r^{n - 1} = 74 \cdot {0.5675675675675675}^{9 - 1} = 74 \cdot {0.5675675675675675}^{8} = 74 \cdot 0.010768136826758694 = 0.7968421251801433$

$a_{10} = a_{1} \cdot r^{n - 1} = 74 \cdot {0.5675675675675675}^{10 - 1} = 74 \cdot {0.5675675675675675}^{9} = 74 \cdot 0.006111645225998177 = 0.4522617467238651$

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几何数列