Aerosol particles released from emission sources undergo various atmospheric and aerosol processes before they become a part of the background aerosols. Coagulation and dispersion, the two important processes that governs evolution of particle characteristics in a puff with an inhomogeneous size distribution of particles in space is considered in this study. This specific case consists of an initial Gaussian aerosol packet in which larger particles are preferentially segregated to farther distances from the centre of the packet. The coagulation-dispersion equation is solved using Jaffe approximation technique for obtaining essential results such as survival fraction, that is, fraction of particles surviving due to the simultaneous action of coagulation and dispersion. Analytical results are developed for the temporal variations of the number concentration, mean size and the standard deviation. The asymptotic results yield the final characteristics of the spectra of the particles which form a part of the background aerosols. These quantities are useful for defining the "effective aerosol source terms" in the general dynamic equations for background aerosols, long-range transport of aerosols, and geo-engineering applications. The results are further discussed.