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解答 - 几何数列

公比是：

r = 0.7636363636363637

r=0.7636363636363637

该系列的和是：

s = 97

s=97

此系列的通用形式是：

a_{n} = 55 \cdot {0.7636363636363637}^{n - 1}

a_n=55*0.7636363636363637^(n-1)

这个序列的第n项是：

55, 42, 32.07272727272727, 24.491900826446283, 18.702906085649893, 14.282219192678099, 10.906421928954185, 8.32854038211047, 6.3599762917934495, 4.856709168278634

55,42,32.07272727272727,24.491900826446283,18.702906085649893,14.282219192678099,10.906421928954185,8.32854038211047,6.3599762917934495,4.856709168278634

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：

$a_{2} a_{1} = \frac{42}{55} = 0.7636363636363637$

该序列的公比（ $r$ ）保持不变，并且等于两个连续项的商。
$r = 0.7636363636363637$

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： $a = 55$ 、公比： $r = 0.7636363636363637$ 和元素数目 $n = 2$ 插入几何级数求和公式：

$s_{2} = 55 * ((1 - {0.7636363636363637}^{2}) / (1 - 0.7636363636363637))$

$s_{2} = 55 * ((1 - 0.5831404958677686) / (1 - 0.7636363636363637))$

$s_{2} = 55 * (0.4168595041322314 / (1 - 0.7636363636363637))$

$s_{2} = 55 * (0.4168595041322314 / 0.23636363636363633)$

$s_{2} = 55 \cdot 1.763636363636364$

$s_{2} = 97.00000000000001$

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： $a = 55$ 和公比： $r = 0.7636363636363637$ 插入几何级数的公式：

$a_{n} = 55 \cdot {0.7636363636363637}^{n - 1}$

4. 找到第n项

使用通用公式找到第n项

$a_{1} = 55$

$a_{2} = a_{1} \cdot r^{n - 1} = 55 \cdot {0.7636363636363637}^{2 - 1} = 55 \cdot {0.7636363636363637}^{1} = 55 \cdot 0.7636363636363637 = 42$

$a_{3} = a_{1} \cdot r^{n - 1} = 55 \cdot {0.7636363636363637}^{3 - 1} = 55 \cdot {0.7636363636363637}^{2} = 55 \cdot 0.5831404958677686 = 32.07272727272727$

$a_{4} = a_{1} \cdot r^{n - 1} = 55 \cdot {0.7636363636363637}^{4 - 1} = 55 \cdot {0.7636363636363637}^{3} = 55 \cdot 0.4453072877535688 = 24.491900826446283$

$a_{5} = a_{1} \cdot r^{n - 1} = 55 \cdot {0.7636363636363637}^{5 - 1} = 55 \cdot {0.7636363636363637}^{4} = 55 \cdot 0.3400528379209071 = 18.702906085649893$

$a_{6} = a_{1} \cdot r^{n - 1} = 55 \cdot {0.7636363636363637}^{6 - 1} = 55 \cdot {0.7636363636363637}^{5} = 55 \cdot 0.25967671259414726 = 14.282219192678099$

$a_{7} = a_{1} \cdot r^{n - 1} = 55 \cdot {0.7636363636363637}^{7 - 1} = 55 \cdot {0.7636363636363637}^{6} = 55 \cdot 0.19829858052643973 = 10.906421928954185$

$a_{8} = a_{1} \cdot r^{n - 1} = 55 \cdot {0.7636363636363637}^{8 - 1} = 55 \cdot {0.7636363636363637}^{7} = 55 \cdot 0.15142800694746308 = 8.32854038211047$

$a_{9} = a_{1} \cdot r^{n - 1} = 55 \cdot {0.7636363636363637}^{9 - 1} = 55 \cdot {0.7636363636363637}^{8} = 55 \cdot 0.11563593257806272 = 6.3599762917934495$

$a_{10} = a_{1} \cdot r^{n - 1} = 55 \cdot {0.7636363636363637}^{10 - 1} = 55 \cdot {0.7636363636363637}^{9} = 55 \cdot 0.08830380305961152 = 4.856709168278634$

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几何数列