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解答 - 几何数列

公比是：

r = 0.7407407407407407

r=0.7407407407407407

该系列的和是：

s = 141

s=141

此系列的通用形式是：

a_{n} = 81 \cdot {0.7407407407407407}^{n - 1}

a_n=81*0.7407407407407407^(n-1)

这个序列的第n项是：

81, 60, 44.444444444444436, 32.92181069958848, 24.386526444139605, 18.064093662325636, 13.38081012024121, 9.911711200178672, 7.342008296428646, 5.4385246640212195

81,60,44.444444444444436,32.92181069958848,24.386526444139605,18.064093662325636,13.38081012024121,9.911711200178672,7.342008296428646,5.4385246640212195

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：

$a_{2} a_{1} = \frac{60}{81} = 0.7407407407407407$

该序列的公比（ $r$ ）保持不变，并且等于两个连续项的商。
$r = 0.7407407407407407$

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： $a = 81$ 、公比： $r = 0.7407407407407407$ 和元素数目 $n = 2$ 插入几何级数求和公式：

$s_{2} = 81 * ((1 - {0.7407407407407407}^{2}) / (1 - 0.7407407407407407))$

$s_{2} = 81 * ((1 - 0.5486968449931412) / (1 - 0.7407407407407407))$

$s_{2} = 81 * (0.4513031550068588 / (1 - 0.7407407407407407))$

$s_{2} = 81 * (0.4513031550068588 / 0.2592592592592593)$

$s_{2} = 81 \cdot 1.740740740740741$

$s_{2} = 141.00000000000003$

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： $a = 81$ 和公比： $r = 0.7407407407407407$ 插入几何级数的公式：

$a_{n} = 81 \cdot {0.7407407407407407}^{n - 1}$

4. 找到第n项

使用通用公式找到第n项

$a_{1} = 81$

$a_{2} = a_{1} \cdot r^{n - 1} = 81 \cdot {0.7407407407407407}^{2 - 1} = 81 \cdot {0.7407407407407407}^{1} = 81 \cdot 0.7407407407407407 = 60$

$a_{3} = a_{1} \cdot r^{n - 1} = 81 \cdot {0.7407407407407407}^{3 - 1} = 81 \cdot {0.7407407407407407}^{2} = 81 \cdot 0.5486968449931412 = 44.444444444444436$

$a_{4} = a_{1} \cdot r^{n - 1} = 81 \cdot {0.7407407407407407}^{4 - 1} = 81 \cdot {0.7407407407407407}^{3} = 81 \cdot 0.40644210740232684 = 32.92181069958848$

$a_{5} = a_{1} \cdot r^{n - 1} = 81 \cdot {0.7407407407407407}^{5 - 1} = 81 \cdot {0.7407407407407407}^{4} = 81 \cdot 0.30106822770542724 = 24.386526444139605$

$a_{6} = a_{1} \cdot r^{n - 1} = 81 \cdot {0.7407407407407407}^{6 - 1} = 81 \cdot {0.7407407407407407}^{5} = 81 \cdot 0.22301350200402018 = 18.064093662325636$

$a_{7} = a_{1} \cdot r^{n - 1} = 81 \cdot {0.7407407407407407}^{7 - 1} = 81 \cdot {0.7407407407407407}^{6} = 81 \cdot 0.16519518666964456 = 13.38081012024121$

$a_{8} = a_{1} \cdot r^{n - 1} = 81 \cdot {0.7407407407407407}^{8 - 1} = 81 \cdot {0.7407407407407407}^{7} = 81 \cdot 0.12236680494047744 = 9.911711200178672$

$a_{9} = a_{1} \cdot r^{n - 1} = 81 \cdot {0.7407407407407407}^{9 - 1} = 81 \cdot {0.7407407407407407}^{8} = 81 \cdot 0.09064207773368699 = 7.342008296428646$

$a_{10} = a_{1} \cdot r^{n - 1} = 81 \cdot {0.7407407407407407}^{10 - 1} = 81 \cdot {0.7407407407407407}^{9} = 81 \cdot 0.0671422798027311 = 5.4385246640212195$

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