formula to divide a distance into X parts with incremental step of Y

Question

I have a mathematical problem that is beyond my ability to solve, so I thought I would ask here:

I have a distance between two values: say 100 and 10. And I have a step value say 5. My need is that I need to reach from 100 to 10 in 5 steps (which is pretty simple). The complexity is that I need an "incremental" step, such that:

step 1 distance < step 2 distance < step 3 distance...so on till the final step. The idea is that each step should be incrementally higher than the previous step. The increment can be something I decide (I dont know how to, thought), but the thing is that there should be a structure to the increment- as in, I cant just randomly keep adding say 10 as first step, 12 as second step, 30 as third step etc.

Can anyone help me formulate something that will help solve this problem I have at hand? Any pointers is greatly appreciated!

Answer 1

There are a couple ways to go about it. The simplest, IMO, would be to let the first step be $x$, the second $x+d$, the third step $x+d+d$, then $x+d+d+d$, and the fifth step $x+d+d+d+d$.

This means you would have to cover the entire distance in $5x+10d$. So you set the distance equal to that. In your example of going from 100 to 10, $$90=5x+10d$$ This gives you multiple ways. For example, $d=5, x=8$ gives steps of $8,13,18,23,28$

Or $d=1,x=16$ yields $16,17,18,19,20$

Answer 2

You can solve it as 10+(a-2d)+(a-d)+a+(a+d)+(a+2d)=100 to get a=18 therefore you need to take steps of value 18 each but that won't solve your problem it seems like but hold on.

You can use 10+(10+d)+(10+2d)+(10+3d)+(10+4d)+(10+5d)=100 to get d=8/3

So you steps would be of length 38/3,46/3,18,62/3,70/3. To confirm this is correct (38+46+54+62+70)3=270/3=90 Hence the solution works.

This answer is a subset of the answer you can find by plotting the graph of 90=5a+10d where a can plotted on the x axis or y and d can be plotted on y axis or x.

Hopes this solves the question

Answer 3

I am not sure if you are familiar with sequences other than the arithmetic sequence, so I hope this helps. The problem you proposed must satisfy the following equations: $$x_0 = 10$$ $$x_1 = x_0 + a_1$$ $$x_2 = x_1 + a_2$$ $$x_3 = x_2 + a_3$$ $$x_4 = x_3 + a_4$$ $$x_5 = x_4 +a_5 = 100$$ $$x_5 = x_0 + a_1 + a_2 + a_3 + a_4 + a_5$$ $$a_1 + a_2 + a_3 + a_4 + a_5 = 90$$ $$a_1 < a_2 < a_3 < a_4 < a_5$$ So we can replace the steps by any strictly increasing sequences which satisfy $a_1 + a_2 + a_3 + a_4 + a_5 = 90$. There are many possible sequences which satisfy this. Here are a few: $$a_i = 15 + i$$ $$a_i = (i+1)^2$$ $$a_i = a + d(i-1) $$ The last one is the example given by turkeyhundt. So if you want to find other step sizes, you just have to make sure $a_1 + a_2 + a_3 + a_4 + a_5 = 90, \ \& \ \ a_1 < a_2 < a_3 < a_4 < a_5$.