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解答 - 几何数列

公比是：

r = 0.8461538461538461

r=0.8461538461538461

该系列的和是：

s = 24

s=24

此系列的通用形式是：

a_{n} = 13 \cdot {0.8461538461538461}^{n - 1}

a_n=13*0.8461538461538461^(n-1)

这个序列的第n项是：

13, 11, 9.307692307692308, 7.875739644970414, 6.664087391898042, 5.638843177759882, 4.7713288427199, 4.037278251532223, 3.4161585205272655, 2.8905956712153786

13,11,9.307692307692308,7.875739644970414,6.664087391898042,5.638843177759882,4.7713288427199,4.037278251532223,3.4161585205272655,2.8905956712153786

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：

$a_{2} a_{1} = \frac{11}{13} = 0.8461538461538461$

该序列的公比（ $r$ ）保持不变，并且等于两个连续项的商。
$r = 0.8461538461538461$

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： $a = 13$ 、公比： $r = 0.8461538461538461$ 和元素数目 $n = 2$ 插入几何级数求和公式：

$s_{2} = 13 * ((1 - {0.8461538461538461}^{2}) / (1 - 0.8461538461538461))$

$s_{2} = 13 * ((1 - 0.7159763313609467) / (1 - 0.8461538461538461))$

$s_{2} = 13 * (0.28402366863905326 / (1 - 0.8461538461538461))$

$s_{2} = 13 * (0.28402366863905326 / 0.15384615384615385)$

$s_{2} = 13 \cdot 1.846153846153846$

$s_{2} = 24$

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： $a = 13$ 和公比： $r = 0.8461538461538461$ 插入几何级数的公式：

$a_{n} = 13 \cdot {0.8461538461538461}^{n - 1}$

4. 找到第n项

使用通用公式找到第n项

$a_{1} = 13$

$a_{2} = a_{1} \cdot r^{n - 1} = 13 \cdot {0.8461538461538461}^{2 - 1} = 13 \cdot {0.8461538461538461}^{1} = 13 \cdot 0.8461538461538461 = 11$

$a_{3} = a_{1} \cdot r^{n - 1} = 13 \cdot {0.8461538461538461}^{3 - 1} = 13 \cdot {0.8461538461538461}^{2} = 13 \cdot 0.7159763313609467 = 9.307692307692308$

$a_{4} = a_{1} \cdot r^{n - 1} = 13 \cdot {0.8461538461538461}^{4 - 1} = 13 \cdot {0.8461538461538461}^{3} = 13 \cdot 0.6058261265361857 = 7.875739644970414$

$a_{5} = a_{1} \cdot r^{n - 1} = 13 \cdot {0.8461538461538461}^{5 - 1} = 13 \cdot {0.8461538461538461}^{4} = 13 \cdot 0.5126221070690802 = 6.664087391898042$

$a_{6} = a_{1} \cdot r^{n - 1} = 13 \cdot {0.8461538461538461}^{6 - 1} = 13 \cdot {0.8461538461538461}^{5} = 13 \cdot 0.43375716751999094 = 5.638843177759882$

$a_{7} = a_{1} \cdot r^{n - 1} = 13 \cdot {0.8461538461538461}^{7 - 1} = 13 \cdot {0.8461538461538461}^{6} = 13 \cdot 0.36702529559383845 = 4.7713288427199$

$a_{8} = a_{1} \cdot r^{n - 1} = 13 \cdot {0.8461538461538461}^{8 - 1} = 13 \cdot {0.8461538461538461}^{7} = 13 \cdot 0.3105598655024787 = 4.037278251532223$

$a_{9} = a_{1} \cdot r^{n - 1} = 13 \cdot {0.8461538461538461}^{9 - 1} = 13 \cdot {0.8461538461538461}^{8} = 13 \cdot 0.2627814246559435 = 3.4161585205272655$

$a_{10} = a_{1} \cdot r^{n - 1} = 13 \cdot {0.8461538461538461}^{10 - 1} = 13 \cdot {0.8461538461538461}^{9} = 13 \cdot 0.22235351317041374 = 2.8905956712153786$

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几何数列