Solving of basic recurrence relation

Question

let us suppose we have following recurrence relation

$T(n)=T(n-1)+1$ where $T_{0}=1$

we need to find homogeneous solution and particular solution , for homogeneous solution, we have

$T(n)-T(n-1)=0$ so in index form $a_n-a_{n-1}=0$ therefore $r-1=0$ , from where we have $r=1$ and general solution will be $a_n=\alpha*r^n$ , because $r=1$ we have $a_n=\alpha$ now for particular solution we have $f(n)=1$ , that why simple take particular solution $T_p=A$ if we put get

$A=A+1$ but how to solve this equation? please help me

Answer 1

The solution to $T(n) = T(n-1) + 1$ is $T(n) = n + T(0)$.
The solution to $T(n) = T(n-1)$ is $T(n) = T(0)$.