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解答 - 几何数列

公比是：

r = 1.8518518518518519

r=1.8518518518518519

该系列的和是：

s = 154

s=154

此系列的通用形式是：

a_{n} = 54 \cdot {1.8518518518518519}^{n - 1}

a_n=54*1.8518518518518519^(n-1)

这个序列的第n项是：

54, 100, 185.1851851851852, 342.9355281207133, 635.0657928161357, 1176.0477644743255, 2177.86623050801, 4033.0856120518706, 7468.677059355316, 13830.883443250586

54,100,185.1851851851852,342.9355281207133,635.0657928161357,1176.0477644743255,2177.86623050801,4033.0856120518706,7468.677059355316,13830.883443250586

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：

$a_{2} a_{1} = \frac{100}{54} = 1.8518518518518519$

该序列的公比（ $r$ ）保持不变，并且等于两个连续项的商。
$r = 1.8518518518518519$

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： $a = 54$ 、公比： $r = 1.8518518518518519$ 和元素数目 $n = 2$ 插入几何级数求和公式：

$s_{2} = 54 * ((1 - {1.8518518518518519}^{2}) / (1 - 1.8518518518518519))$

$s_{2} = 54 * ((1 - 3.4293552812071333) / (1 - 1.8518518518518519))$

$s_{2} = 54 * (- 2.4293552812071333 / (1 - 1.8518518518518519))$

$s_{2} = 54 * (- 2.4293552812071333 / - 0.8518518518518519)$

$s_{2} = 54 \cdot 2.851851851851852$

$s_{2} = 154$

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： $a = 54$ 和公比： $r = 1.8518518518518519$ 插入几何级数的公式：

$a_{n} = 54 \cdot {1.8518518518518519}^{n - 1}$

4. 找到第n项

使用通用公式找到第n项

$a_{1} = 54$

$a_{2} = a_{1} \cdot r^{n - 1} = 54 \cdot {1.8518518518518519}^{2 - 1} = 54 \cdot {1.8518518518518519}^{1} = 54 \cdot 1.8518518518518519 = 100$

$a_{3} = a_{1} \cdot r^{n - 1} = 54 \cdot {1.8518518518518519}^{3 - 1} = 54 \cdot {1.8518518518518519}^{2} = 54 \cdot 3.4293552812071333 = 185.1851851851852$

$a_{4} = a_{1} \cdot r^{n - 1} = 54 \cdot {1.8518518518518519}^{4 - 1} = 54 \cdot {1.8518518518518519}^{3} = 54 \cdot 6.3506579281613575 = 342.9355281207133$

$a_{5} = a_{1} \cdot r^{n - 1} = 54 \cdot {1.8518518518518519}^{5 - 1} = 54 \cdot {1.8518518518518519}^{4} = 54 \cdot 11.760477644743254 = 635.0657928161357$

$a_{6} = a_{1} \cdot r^{n - 1} = 54 \cdot {1.8518518518518519}^{6 - 1} = 54 \cdot {1.8518518518518519}^{5} = 54 \cdot 21.7786623050801 = 1176.0477644743255$

$a_{7} = a_{1} \cdot r^{n - 1} = 54 \cdot {1.8518518518518519}^{7 - 1} = 54 \cdot {1.8518518518518519}^{6} = 54 \cdot 40.33085612051871 = 2177.86623050801$

$a_{8} = a_{1} \cdot r^{n - 1} = 54 \cdot {1.8518518518518519}^{8 - 1} = 54 \cdot {1.8518518518518519}^{7} = 54 \cdot 74.68677059355316 = 4033.0856120518706$

$a_{9} = a_{1} \cdot r^{n - 1} = 54 \cdot {1.8518518518518519}^{9 - 1} = 54 \cdot {1.8518518518518519}^{8} = 54 \cdot 138.30883443250585 = 7468.677059355316$

$a_{10} = a_{1} \cdot r^{n - 1} = 54 \cdot {1.8518518518518519}^{10 - 1} = 54 \cdot {1.8518518518518519}^{9} = 54 \cdot 256.12747117130715 = 13830.883443250586$

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