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解答 - 几何数列

公比是：

r = 0.6226415094339622

r=0.6226415094339622

该系列的和是：

s = 258

s=258

此系列的通用形式是：

a_{n} = 159 \cdot {0.6226415094339622}^{n - 1}

a_n=159*0.6226415094339622^(n-1)

这个序列的第n项是：

159, 99, 61.64150943396226, 38.38056247775008, 23.897331354070804, 14.8794704657422, 9.264575950367783, 5.7685095540025815, 3.5917134958884, 2.236349912534287

159,99,61.64150943396226,38.38056247775008,23.897331354070804,14.8794704657422,9.264575950367783,5.7685095540025815,3.5917134958884,2.236349912534287

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：

$a_{2} a_{1} = \frac{99}{159} = 0.6226415094339622$

该序列的公比（ $r$ ）保持不变，并且等于两个连续项的商。
$r = 0.6226415094339622$

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： $a = 159$ 、公比： $r = 0.6226415094339622$ 和元素数目 $n = 2$ 插入几何级数求和公式：

$s_{2} = 159 * ((1 - {0.6226415094339622}^{2}) / (1 - 0.6226415094339622))$

$s_{2} = 159 * ((1 - 0.3876824492702029) / (1 - 0.6226415094339622))$

$s_{2} = 159 * (0.6123175507297971 / (1 - 0.6226415094339622))$

$s_{2} = 159 * (0.6123175507297971 / 0.37735849056603776)$

$s_{2} = 159 \cdot 1.6226415094339621$

$s_{2} = 258$

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： $a = 159$ 和公比： $r = 0.6226415094339622$ 插入几何级数的公式：

$a_{n} = 159 \cdot {0.6226415094339622}^{n - 1}$

4. 找到第n项

使用通用公式找到第n项

$a_{1} = 159$

$a_{2} = a_{1} \cdot r^{n - 1} = 159 \cdot {0.6226415094339622}^{2 - 1} = 159 \cdot {0.6226415094339622}^{1} = 159 \cdot 0.6226415094339622 = 99$

$a_{3} = a_{1} \cdot r^{n - 1} = 159 \cdot {0.6226415094339622}^{3 - 1} = 159 \cdot {0.6226415094339622}^{2} = 159 \cdot 0.3876824492702029 = 61.64150943396226$

$a_{4} = a_{1} \cdot r^{n - 1} = 159 \cdot {0.6226415094339622}^{4 - 1} = 159 \cdot {0.6226415094339622}^{3} = 159 \cdot 0.24138718539465462 = 38.38056247775008$

$a_{5} = a_{1} \cdot r^{n - 1} = 159 \cdot {0.6226415094339622}^{5 - 1} = 159 \cdot {0.6226415094339622}^{4} = 159 \cdot 0.15029768147214342 = 23.897331354070804$

$a_{6} = a_{1} \cdot r^{n - 1} = 159 \cdot {0.6226415094339622}^{6 - 1} = 159 \cdot {0.6226415094339622}^{5} = 159 \cdot 0.09358157525624025 = 14.8794704657422$

$a_{7} = a_{1} \cdot r^{n - 1} = 159 \cdot {0.6226415094339622}^{7 - 1} = 159 \cdot {0.6226415094339622}^{6} = 159 \cdot 0.058267773272753356 = 9.264575950367783$

$a_{8} = a_{1} \cdot r^{n - 1} = 159 \cdot {0.6226415094339622}^{8 - 1} = 159 \cdot {0.6226415094339622}^{7} = 159 \cdot 0.03627993430190303 = 5.7685095540025815$

$a_{9} = a_{1} \cdot r^{n - 1} = 159 \cdot {0.6226415094339622}^{9 - 1} = 159 \cdot {0.6226415094339622}^{8} = 159 \cdot 0.022589393055901887 = 3.5917134958884$

$a_{10} = a_{1} \cdot r^{n - 1} = 159 \cdot {0.6226415094339622}^{10 - 1} = 159 \cdot {0.6226415094339622}^{9} = 159 \cdot 0.014065093789523817 = 2.236349912534287$

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