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解答 - 几何数列

公比是：

r = 0.45977011494252873

r=0.45977011494252873

该系列的和是：

s = 126

s=126

此系列的通用形式是：

a_{n} = 87 \cdot {0.45977011494252873}^{n - 1}

a_n=87*0.45977011494252873^(n-1)

这个序列的第n项是：

87, 40, 18.39080459770115, 8.455542343770643, 3.8876056752968475, 1.7874049081824586, 0.821795360083889, 0.37783694716500643, 0.17371813662758917, 0.07987040764486858

87,40,18.39080459770115,8.455542343770643,3.8876056752968475,1.7874049081824586,0.821795360083889,0.37783694716500643,0.17371813662758917,0.07987040764486858

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：

$a_{2} a_{1} = \frac{40}{87} = 0.45977011494252873$

该序列的公比（ $r$ ）保持不变，并且等于两个连续项的商。
$r = 0.45977011494252873$

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： $a = 87$ 、公比： $r = 0.45977011494252873$ 和元素数目 $n = 2$ 插入几何级数求和公式：

$s_{2} = 87 * ((1 - {0.45977011494252873}^{2}) / (1 - 0.45977011494252873))$

$s_{2} = 87 * ((1 - 0.21138855859426608) / (1 - 0.45977011494252873))$

$s_{2} = 87 * (0.7886114414057339 / (1 - 0.45977011494252873))$

$s_{2} = 87 * (0.7886114414057339 / 0.5402298850574713)$

$s_{2} = 87 \cdot 1.4597701149425286$

$s_{2} = 126.99999999999999$

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： $a = 87$ 和公比： $r = 0.45977011494252873$ 插入几何级数的公式：

$a_{n} = 87 \cdot {0.45977011494252873}^{n - 1}$

4. 找到第n项

使用通用公式找到第n项

$a_{1} = 87$

$a_{2} = a_{1} \cdot r^{n - 1} = 87 \cdot {0.45977011494252873}^{2 - 1} = 87 \cdot {0.45977011494252873}^{1} = 87 \cdot 0.45977011494252873 = 40$

$a_{3} = a_{1} \cdot r^{n - 1} = 87 \cdot {0.45977011494252873}^{3 - 1} = 87 \cdot {0.45977011494252873}^{2} = 87 \cdot 0.21138855859426608 = 18.39080459770115$

$a_{4} = a_{1} \cdot r^{n - 1} = 87 \cdot {0.45977011494252873}^{4 - 1} = 87 \cdot {0.45977011494252873}^{3} = 87 \cdot 0.09719014188242119 = 8.455542343770643$

$a_{5} = a_{1} \cdot r^{n - 1} = 87 \cdot {0.45977011494252873}^{5 - 1} = 87 \cdot {0.45977011494252873}^{4} = 87 \cdot 0.044685122704561465 = 3.8876056752968475$

$a_{6} = a_{1} \cdot r^{n - 1} = 87 \cdot {0.45977011494252873}^{6 - 1} = 87 \cdot {0.45977011494252873}^{5} = 87 \cdot 0.020544884002097225 = 1.7874049081824586$

$a_{7} = a_{1} \cdot r^{n - 1} = 87 \cdot {0.45977011494252873}^{7 - 1} = 87 \cdot {0.45977011494252873}^{6} = 87 \cdot 0.00944592367912516 = 0.821795360083889$

$a_{8} = a_{1} \cdot r^{n - 1} = 87 \cdot {0.45977011494252873}^{8 - 1} = 87 \cdot {0.45977011494252873}^{7} = 87 \cdot 0.004342953415689729 = 0.37783694716500643$

$a_{9} = a_{1} \cdot r^{n - 1} = 87 \cdot {0.45977011494252873}^{9 - 1} = 87 \cdot {0.45977011494252873}^{8} = 87 \cdot 0.0019967601911217146 = 0.17371813662758917$

$a_{10} = a_{1} \cdot r^{n - 1} = 87 \cdot {0.45977011494252873}^{10 - 1} = 87 \cdot {0.45977011494252873}^{9} = 87 \cdot 0.0009180506625846963 = 0.07987040764486858$

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