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解答 - 几何数列

公比是：

r = 0.9245283018867925

r=0.9245283018867925

该系列的和是：

s = 101

s=101

此系列的通用形式是：

a_{n} = 53 \cdot {0.9245283018867925}^{n - 1}

a_n=53*0.9245283018867925^(n-1)

这个序列的第n项是：

53, 49, 45.301886792452834, 41.882876468494125, 38.72190465955118, 35.799496760717126, 33.097647948587536, 30.59971225435451, 28.29030000874285, 26.155183026950937

53,49,45.301886792452834,41.882876468494125,38.72190465955118,35.799496760717126,33.097647948587536,30.59971225435451,28.29030000874285,26.155183026950937

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：

$a_{2} a_{1} = \frac{49}{53} = 0.9245283018867925$

该序列的公比（ $r$ ）保持不变，并且等于两个连续项的商。
$r = 0.9245283018867925$

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： $a = 53$ 、公比： $r = 0.9245283018867925$ 和元素数目 $n = 2$ 插入几何级数求和公式：

$s_{2} = 53 * ((1 - {0.9245283018867925}^{2}) / (1 - 0.9245283018867925))$

$s_{2} = 53 * ((1 - 0.8547525809896761) / (1 - 0.9245283018867925))$

$s_{2} = 53 * (0.1452474190103239 / (1 - 0.9245283018867925))$

$s_{2} = 53 * (0.1452474190103239 / 0.07547169811320753)$

$s_{2} = 53 \cdot 1.924528301886792$

$s_{2} = 101.99999999999997$

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： $a = 53$ 和公比： $r = 0.9245283018867925$ 插入几何级数的公式：

$a_{n} = 53 \cdot {0.9245283018867925}^{n - 1}$

4. 找到第n项

使用通用公式找到第n项

$a_{1} = 53$

$a_{2} = a_{1} \cdot r^{n - 1} = 53 \cdot {0.9245283018867925}^{2 - 1} = 53 \cdot {0.9245283018867925}^{1} = 53 \cdot 0.9245283018867925 = 49$

$a_{3} = a_{1} \cdot r^{n - 1} = 53 \cdot {0.9245283018867925}^{3 - 1} = 53 \cdot {0.9245283018867925}^{2} = 53 \cdot 0.8547525809896761 = 45.301886792452834$

$a_{4} = a_{1} \cdot r^{n - 1} = 53 \cdot {0.9245283018867925}^{4 - 1} = 53 \cdot {0.9245283018867925}^{3} = 53 \cdot 0.7902429522357383 = 41.882876468494125$

$a_{5} = a_{1} \cdot r^{n - 1} = 53 \cdot {0.9245283018867925}^{5 - 1} = 53 \cdot {0.9245283018867925}^{4} = 53 \cdot 0.7306019747085127 = 38.72190465955118$

$a_{6} = a_{1} \cdot r^{n - 1} = 53 \cdot {0.9245283018867925}^{6 - 1} = 53 \cdot {0.9245283018867925}^{5} = 53 \cdot 0.6754622030323986 = 35.799496760717126$

$a_{7} = a_{1} \cdot r^{n - 1} = 53 \cdot {0.9245283018867925}^{7 - 1} = 53 \cdot {0.9245283018867925}^{6} = 53 \cdot 0.6244839235582553 = 33.097647948587536$

$a_{8} = a_{1} \cdot r^{n - 1} = 53 \cdot {0.9245283018867925}^{8 - 1} = 53 \cdot {0.9245283018867925}^{7} = 53 \cdot 0.5773530614029153 = 30.59971225435451$

$a_{9} = a_{1} \cdot r^{n - 1} = 53 \cdot {0.9245283018867925}^{9 - 1} = 53 \cdot {0.9245283018867925}^{8} = 53 \cdot 0.5337792454479783 = 28.29030000874285$

$a_{10} = a_{1} \cdot r^{n - 1} = 53 \cdot {0.9245283018867925}^{10 - 1} = 53 \cdot {0.9245283018867925}^{9} = 53 \cdot 0.4934940193764328 = 26.155183026950937$

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