If I have a recursive sequence defined by:
$a_0 = 7$
$a_n = a_{n-1} + 3 + 2(n-1),$ for $n \geq 1$
How is this recursive sequence the same as the one above. Isn’t $n+1 \geq 2$?
$a_0 = 7$
$a_{n+1} = a_{n} + 3 + 2(n)$, for $n \geq 0$
How do we get from the top recursive sequence to the one on the bottom? Thank you!
One suggestion is to define $n'=n-1$ and make the adjustments as appropriate.
In this notation, the first recursion becomes:
$$ a_{n'+1}=a_{n'}+3+2\left(n'\right) $$
This expression is given as being valid for $n \ge 1$, which translates to $n'+1 \ge 1$ or $n' \ge 0.$
This shows the transformation from the first to the second recursion.
