Question

How do you write a recursive and explicit equation for Arithmetic sequences?

Answer 1

A recursive equation has the "unknown" on the right hand side.
For example,
the sequence is 1,2,3,4,5,6,....
and \(T_1=1\).
Then we can say that \(T_n=T_{n-1}\), given that \(T_1=1\) .
A recursive equation must provide a boundary condition (for example, \(T_1=1\) , for the sequence to start (or end).
A famous example of a recursive sequence is the fibonacci numbers, where
\(F_n=F_{n-1}+F_{n-2}\) given \(F_0=0,~and~ F_1=1\)
Note that the Fibonacci numbers does NOT form an arithmetic sequence.

An explicit equation tells you exactly what the value is without knowing previous terms.
Example:
sequence 1,2,3,4,5...
\(T_n = n\)
While the explicit equation for the Fibonacci sequence is
\(\large F_n=\frac{(1+\sqrt 5)^n-(1-\sqrt 5)^n}{2^n \sqrt 5}\)

This way, you see how recurrent equations could be simpler, but takes more efforts to solve.

Answer 2

Then we can say that \(Tn=T_{n−1}+1\), given that T1=1 .

Answer 3

So if t1=1, t2=t1+1=2, t3=t2+1=3....
Since we are using a previous term to calculate the present term, this is called recursion.