  In this paper, we introduce a novel indefinite summation $\sum_{t} f(t)$ (or
antidifference $\Delta ^{-1}f(t) $ ) formula for any given function $f$. We
apply the indefinite summation formula to calculate a particular solution to a
nonhomogeneous linear difference equation of the form $$ y(x+h)-\lambda
y(x)=f(x),\quad h > 0,\quad \lambda \neq 0, $$ and also to solve a linear
difference inequality of the form $$ y(x+h)-\lambda y(x) \geq 0,\quad h >
0,\quad \lambda \neq 0. $$ Furthermore, we apply the formula to determine a
particular solution to a difference equations of the form $$ \Phi(E)y(t)=f(t),
$$ and in solving a linear difference inequality of the form, $$
\Phi(E)y(t)\geq 0, $$ where $ \Phi(E) $ is some linear difference operator. We
show how the antidifference of a function $f$ calculated with the current
formula is related to the already existing result and establish the
corresponding identity.
