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解答 - 几何数列

公比是：

r = 0.5070707070707071

r=0.5070707070707071

该系列的和是：

s = 1492

s=1492

此系列的通用形式是：

a_{n} = 990 \cdot {0.5070707070707071}^{n - 1}

a_n=990*0.5070707070707071^(n-1)

这个序列的第n项是：

990, 502.00000000000006, 254.54949494949497, 129.0745923885318, 65.44994482731613, 33.187749801325964, 16.828535757844076, 8.533257525694674, 4.3269649271704305, 2.1940771650904614

990,502.00000000000006,254.54949494949497,129.0745923885318,65.44994482731613,33.187749801325964,16.828535757844076,8.533257525694674,4.3269649271704305,2.1940771650904614

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：

$a_{2} a_{1} = \frac{502}{990} = 0.5070707070707071$

该序列的公比（ $r$ ）保持不变，并且等于两个连续项的商。
$r = 0.5070707070707071$

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： $a = 990$ 、公比： $r = 0.5070707070707071$ 和元素数目 $n = 2$ 插入几何级数求和公式：

$s_{2} = 990 * ((1 - {0.5070707070707071}^{2}) / (1 - 0.5070707070707071))$

$s_{2} = 990 * ((1 - 0.25712070196918685) / (1 - 0.5070707070707071))$

$s_{2} = 990 * (0.7428792980308132 / (1 - 0.5070707070707071))$

$s_{2} = 990 * (0.7428792980308132 / 0.4929292929292929)$

$s_{2} = 990 \cdot 1.5070707070707072$

$s_{2} = 1492.0000000000002$

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： $a = 990$ 和公比： $r = 0.5070707070707071$ 插入几何级数的公式：

$a_{n} = 990 \cdot {0.5070707070707071}^{n - 1}$

4. 找到第n项

使用通用公式找到第n项

$a_{1} = 990$

$a_{2} = a_{1} \cdot r^{n - 1} = 990 \cdot {0.5070707070707071}^{2 - 1} = 990 \cdot {0.5070707070707071}^{1} = 990 \cdot 0.5070707070707071 = 502.00000000000006$

$a_{3} = a_{1} \cdot r^{n - 1} = 990 \cdot {0.5070707070707071}^{3 - 1} = 990 \cdot {0.5070707070707071}^{2} = 990 \cdot 0.25712070196918685 = 254.54949494949497$

$a_{4} = a_{1} \cdot r^{n - 1} = 990 \cdot {0.5070707070707071}^{4 - 1} = 990 \cdot {0.5070707070707071}^{3} = 990 \cdot 0.13037837615003214 = 129.0745923885318$

$a_{5} = a_{1} \cdot r^{n - 1} = 990 \cdot {0.5070707070707071}^{5 - 1} = 990 \cdot {0.5070707070707071}^{4} = 990 \cdot 0.06611105538112741 = 65.44994482731613$

$a_{6} = a_{1} \cdot r^{n - 1} = 990 \cdot {0.5070707070707071}^{6 - 1} = 990 \cdot {0.5070707070707071}^{5} = 990 \cdot 0.033522979597298955 = 33.187749801325964$

$a_{7} = a_{1} \cdot r^{n - 1} = 990 \cdot {0.5070707070707071}^{7 - 1} = 990 \cdot {0.5070707070707071}^{6} = 990 \cdot 0.01699852096751927 = 16.828535757844076$

$a_{8} = a_{1} \cdot r^{n - 1} = 990 \cdot {0.5070707070707071}^{8 - 1} = 990 \cdot {0.5070707070707071}^{7} = 990 \cdot 0.008619452046156236 = 8.533257525694674$

$a_{9} = a_{1} \cdot r^{n - 1} = 990 \cdot {0.5070707070707071}^{9 - 1} = 990 \cdot {0.5070707070707071}^{8} = 990 \cdot 0.004370671643606496 = 4.3269649271704305$

$a_{10} = a_{1} \cdot r^{n - 1} = 990 \cdot {0.5070707070707071}^{10 - 1} = 990 \cdot {0.5070707070707071}^{9} = 990 \cdot 0.0022162395606974356 = 2.1940771650904614$

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几何数列