Tell me why the recursive sequence formula is tn=tn-1+(2n+1)

why, from this set of numbers:
tn 1, 4, 9, 16
n 1, 2, 3, 4

that we use that formula? I know how to find the nth term (just plug in), I just don't understand how someone would know the right strategy (or have the right reasoning) to understand that tn-1+(2n+1) is the formula to solve the nth term? I'm basically asking you to prove the reason for this formula... ?

Here... hope this explanation works...
difference between 1 and 4 = 3
difference between 4 and 9 = 5
difference between 9 and 16 = 7
....
So you can see that the difference between any two consecutive numbers is a odd number...

Now think about how you can make any number .... just any number ..odd?

If you multiply it by 2 ...we know it will be even for sure but if we add just one 1 or subtract one 1... it will automatically be odd
thus they have added 2n+ 1

So basically... any number in a sequence is the sum of previous number and an odd number
1 when n =0
4 = 1 +3 = 1+ (2*1 +1) when n=1
9 = 4+5 = 4 + (2*2 +1) when n=2
16=9+7 = 9 + (2*3 +1) when n=3

n = 1, 2, 3, 4
tn = 1, 4, 9, 16

Thus, tn = n^2, t(n-1) = (n-1)^2, and since

tn - t(n-1) = n^2 - (n-1)^2,

we have

tn = t(n-1) + n^2 - (n-1)^2
tn = t(n-1) + n^2 - n^2 + 2n - 1
= t(n-1) + 2n - 1