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We present a closed-form formula for the general solution to the difference equation $$x_{n+k}-q_{n}x_{n}=f_{n},quad nin mathbb {N}_{0}, $$where (kin mathbb {N}), ((q_{n})_{nin mathbb {N}_{0}}), ((f_{n})_{nin mathbb {N}_{0}}subset mathbb {C}), in the case (q_{n}=q), (nin mathbb {N}_{0}), (qin mathbb {C}setminus{0}). Using the formula, we show the existence of a unique bounded solution to the equation when (|q|u003e1) and (sup_{nin mathbb {N}_{0}}|f_{n}|u003cinfty) by finding a solution in closed form. By using the formula for the bounded solution we introduce an operator that, together with the contraction mapping principle, helps in showing the existence of a unique bounded solution to the equation in the case where the sequence ((q_{n})_{nin mathbb {N}_{0}}) is real and nonconstant, which shows that, in this case, there is an elegant method of proving the result in a unified way. We also obtain some interesting formulas.

General solution to a higher-order linear difference equation and existence of bounded solutions