What is the mathematical formula for increasing a random smaller number by a cer

Question

What is the mathematical formula for increasing a random smaller number by a certain amount of times and stop at another random larger number? For example: If my small number is 10 and my large number is 100 and I want to increase my small number 10 times to get to my large number. This would look like this: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100 If used 1000 as my starting small number and wanted to go to 100000 in 10 steps it would be 10000, 20000, 30000, 40000, 50000, 60000, 70000, 80000, 90000, 100000 How could I demonstrate this mathematically using a formula for each of the intervals? Where no matter what 3 values (small number, large number and intervals) I use it will work.

Explanation / Answer

Formula of arithmetic progression can be used here..

If the first term of the sequence is a and the common difference between each term is d then the arithmetic progression is a, a+d, a+2d, a+3d, ...

where the n-th term is a +(n1)d.

So for your question 10,20,30,40....100 or 10000,20000,30000....100000 or anything as such we will be using the above formula for 10 steps

Like 10,20,30...100

a=10. d=a (which is a itself so you could write d=a here) n=1 to 10

an = a+(n-1)d

n=1 , a1= 10 +(1-1)*10 = 10

n=2 , a2= 10 +(2-1)*10 = 20

n=3 , a3= 10 +(3-1)*10 = 30

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n=10 , a10= 10 +(10-1)*10 = 100

Similarly we can calculate any random number with intervals as common difference and initial value based on any random number in 10 steps (n=1 to 10) to get every value in the series.

You can also write this formula as an= a+(n-1)*a since d=a only so it will change into an= a*n where n = 1 to 10 and an denote the nth term.

Hence your formula in simple terms is  an= a*n where n = 1 to 10 and an denote the nth term.

Note :The above formula an =a*n (n=1 to 10) will work for any series in which terms are multiple of initial value like for series 100 ,200,300... 1000.. we have 1*100 ,2*100... 10*100. But if you have different intervals you can use arithmetic progression formula only with d as an interval i.e an =a+(n-1)*d (n=1 to 10) and initialize d with the given interval( like 6,11,16....51 it has interval 5 and initial value 6)

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