Excel中指数求和的倒数 [英] reverse of exponential sum in Excel

问题描述

我试图在Excel中找到一种从指数值之和算出基数的方法.

I'm trying to find a way, in Excel, to work out the base number from the sum of exponential values.

例如，可能存在:

First number : 10
10 *  1.5 = 15
15 * 1.5 = 22.5
22.5 * 1.5 = 33.75
when you add these together you get 81.25

(它并不总是1.5，并且可能乘以3倍以上)

(It won't always be 1.5 and there may be more than 3 times it's multiplied)

我正在尝试制定一个公式，该公式将从指数值(81.25)的总和中得出第一个数字(10).

I'm trying to work out a formula that will derive the first number (10) from the sum of the exponential values (81.25).

在此方面，我将不胜感激！

I'd appreciate any help at all on this!

推荐答案

您具有几何级数，其中

 a = 10
 q = 1.5
 n = 4 (you have 4 terms: 10, 15, 22.5 and 33.75)

因此，总和 s (这里 ^ 代表提高权力)是

And so the sum s (here ^ stands for raising into power) is

 s = a * (1 - q^n) / (1 - q)

以您的情况

 s = 81.25

要获取 a ，您可以

 a = s * (1 - q) / (1 - q^n) 

编辑:如果我理解您的最后一条评论正确，则"...... 第一乘数可以不同，但​​所有其他都是一样..."你有

If I understand your last comment right "...first multiplier can be different, but all the others are the same..." you have

a
a * b
a * b * q
a * b * q^2
...
a * b * q^(n-2)

术语，然后再次是几何级数:

s = a + a * b * (1 + q + q^2 + ... + q^(n-2)) =
  = a + a * b * (1 - q^(n - 1)) / (1 - q)

如果我们解决 a ，我们将拥有

If we solve for a we'll have

a = s * (1 - q) / ((1 - q) + b * (1 - q^(n - 1)))   

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