Need help with either writing a loop in excel or help with finding a sequential numbers equation.

Question

Ive been stuck on this problem for 2 going on 3 days now and cant find anything that fits my problem. I've done some research and tried many different formulas/equations but can't quite get the right answer. I'm trying to find an equation that will produce these numbers 1=25000, 2=125000, 3=350000, 4=750000, 5=1375000, 6=2275000 and beyond. I have this formula to produce the number I have. =(X)^2*25000 Which are 1=25000,2=100000,3=225000,etc. Trying to make a formula for excel so it adds up all the totals of =(1)^2*25000, =(2)^2*25000, =(3)^2*25000, which makes 25000+100000+225000=350000. when I type in 3. Don't know how to do that in excel so was wondering if there is a formula for it with the product answers as listed above or maybe write a for loop or something. Hope someone has an answer thanks!

Answer 1

Dividing all your numbers by $1000$, consider the sequence $$\{25,125,350,750,1375,2275\}$$ and generate the successive difference tables (have a look here and here ) to get successively $$\{100,225,400,625,900\}$$ $$\{125,175,225,275\}$$ $$\{50,50,50\}$$ $$\{0,0\}$$ So, it is clear that the original sequence corresponds to a cubic equation.

Now, if you are lazy (this is not a sin), use Excel and generate the trend line for a cubic polynomial and try to rationalize the coefficients.

Being less lazy, consider $$f(x)=a x^3+b x^2+c x+d$$ and use the first four data points to write $$a+b+c+d=25000$$ $$8a+4b+2c+d=125000$$ $$27a+9b+3c+d=350000$$ $$64a+16b+4c+d=750000$$ Solve for $a,b,c,d$ using any method of your choice and you will get nice numbers.

The $100^{\text{th}}$ number of the sequence would be $8 $ $458 $ $750 $ $000$.

Your turn now !