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解答 - 几何数列

公比是：

r = 0.5384615384615384

r=0.5384615384615384

该系列的和是：

s = 1000

s=1000

此系列的通用形式是：

a_{n} = 650 \cdot {0.5384615384615384}^{n - 1}

a_n=650*0.5384615384615384^(n-1)

这个序列的第n项是：

650, 350, 188.46153846153845, 101.47928994082838, 54.64269458352298, 29.422989391127754, 15.843148133684174, 8.530925918137632, 4.593575494381802, 2.473463727744047

650,350,188.46153846153845,101.47928994082838,54.64269458352298,29.422989391127754,15.843148133684174,8.530925918137632,4.593575494381802,2.473463727744047

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：

$a_{2} a_{1} = \frac{350}{650} = 0.5384615384615384$

该序列的公比（ $r$ ）保持不变，并且等于两个连续项的商。
$r = 0.5384615384615384$

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： $a = 650$ 、公比： $r = 0.5384615384615384$ 和元素数目 $n = 2$ 插入几何级数求和公式：

$s_{2} = 650 * ((1 - {0.5384615384615384}^{2}) / (1 - 0.5384615384615384))$

$s_{2} = 650 * ((1 - 0.28994082840236685) / (1 - 0.5384615384615384))$

$s_{2} = 650 * (0.7100591715976332 / (1 - 0.5384615384615384))$

$s_{2} = 650 * (0.7100591715976332 / 0.46153846153846156)$

$s_{2} = 650 \cdot 1.5384615384615385$

$s_{2} = 1000$

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： $a = 650$ 和公比： $r = 0.5384615384615384$ 插入几何级数的公式：

$a_{n} = 650 \cdot {0.5384615384615384}^{n - 1}$

4. 找到第n项

使用通用公式找到第n项

$a_{1} = 650$

$a_{2} = a_{1} \cdot r^{n - 1} = 650 \cdot {0.5384615384615384}^{2 - 1} = 650 \cdot {0.5384615384615384}^{1} = 650 \cdot 0.5384615384615384 = 350$

$a_{3} = a_{1} \cdot r^{n - 1} = 650 \cdot {0.5384615384615384}^{3 - 1} = 650 \cdot {0.5384615384615384}^{2} = 650 \cdot 0.28994082840236685 = 188.46153846153845$

$a_{4} = a_{1} \cdot r^{n - 1} = 650 \cdot {0.5384615384615384}^{4 - 1} = 650 \cdot {0.5384615384615384}^{3} = 650 \cdot 0.15612198452435136 = 101.47928994082838$

$a_{5} = a_{1} \cdot r^{n - 1} = 650 \cdot {0.5384615384615384}^{5 - 1} = 650 \cdot {0.5384615384615384}^{4} = 650 \cdot 0.08406568397465074 = 54.64269458352298$

$a_{6} = a_{1} \cdot r^{n - 1} = 650 \cdot {0.5384615384615384}^{6 - 1} = 650 \cdot {0.5384615384615384}^{5} = 650 \cdot 0.04526613752481193 = 29.422989391127754$

$a_{7} = a_{1} \cdot r^{n - 1} = 650 \cdot {0.5384615384615384}^{7 - 1} = 650 \cdot {0.5384615384615384}^{6} = 650 \cdot 0.024374074051821806 = 15.843148133684174$

$a_{8} = a_{1} \cdot r^{n - 1} = 650 \cdot {0.5384615384615384}^{8 - 1} = 650 \cdot {0.5384615384615384}^{7} = 650 \cdot 0.013124501412519434 = 8.530925918137632$

$a_{9} = a_{1} \cdot r^{n - 1} = 650 \cdot {0.5384615384615384}^{9 - 1} = 650 \cdot {0.5384615384615384}^{8} = 650 \cdot 0.007067039222125849 = 4.593575494381802$

$a_{10} = a_{1} \cdot r^{n - 1} = 650 \cdot {0.5384615384615384}^{10 - 1} = 650 \cdot {0.5384615384615384}^{9} = 650 \cdot 0.0038053288119139182 = 2.473463727744047$

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几何数列