# Does solving a recurrence relation by iteration have two different meanings?

I've seen iteration used by plugging numbers in and not simplifying and guessing the explicit formula, e.g., $t_n$ plug $n=1,2,3,4$ in and guess the explicit formula. The other way I've seen is plugging the variables in e.g. plug $t_{n+1}$ in for $t_n$ and try to see the explicit formula that way. Am I right that both these methods are known as iteration? How do you decide which one to use?

For example this is from a textbook:

Let $a_k=a_{k-1}+2$ and $a_0=1$ Use iteration to guess an explicit formula for the sequence. $a_1=a_0+2=1+2$
$a_2=a_1+2=1+2+2$
$a_3=a_2+2=1+2+2+2$
$a_4=a_3+2=1+2+2+2+2$
It appears helpful to use shorthand $a_1=a_0+2=1+2$
$a_2=a_1+2=1+2 \cdot 2$
$a_3=a_2+2=1+3 \cdot 2$
$a_4=a_3+2=1+ 4 \cdot 2$

Guess: $a_n=1+n \cdot 2 = 1+2n$

vs this one

Here is an example of solving the above recurrence relation for g(n) using the iteration method:

  g(n) = g(n-1) + 2n - 1
= [g(n-2) + 2(n-1) - 1] + 2n - 1
// because g(n-1) = g(n-2) + 2(n-1) -1 //
= g(n-2) + 2(n-1) + 2n - 2
= [g(n-3) + 2(n-2) -1] + 2(n-1) + 2n - 2
// because g(n-2) = g(n-3) + 2(n-2) -1 //
= g(n-3) + 2(n-2) + 2(n-1) + 2n - 3
...
= g(n-i) + 2(n-i+1) +...+ 2n - i
...
= g(n-n) + 2(n-n+1) +...+ 2n - n
= 0 + 2 + 4 +...+ 2n - n
// because g(0) = 0 //
= 2 + 4 +...+ 2n - n
= 2*n*(n+1)/2 - n
// using arithmetic progression formula 1+...+n = n(n+1)/2 //
= n^2


Clearly these methods are different so are they both called iteration and when do you know which to use? I guess the first one is simpler since it has fewer variables.