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The geometrically frustrated spin-1 Blume-Capel model on an infinite sawtooth chain is exactly solved by the transfer matrix method. The magnetization, ground-state phase diagram, magnetocaloric properties, and specific heat of the system are investigated. The results indicate that: (i) Magnetization plateaus appear at zero temperature. Their number depends on the sign of the crystal field D. For D≥0 there are two magnetization plateaus; however, for D<0 five plateaus exist. At a finite temperature, thermal excitation will destroy these plateaus completely. (ii) Phase transition between any two long-range-ordered ground states, whose spin configurations are given in phase diagram, is the first-order one. The macroscopic degeneracy of the ground states described by the entropy only exists at phase coexistence points. (iii) As temperature approaches zero, magnetocaloric properties and entropy change sharply near phase coexistence points. (iv) The crossovers of the specific heat from a single-peak structure to double-peak ones can signal the phase coexistence points in ground-state phase diagram. [ABSTRACT FROM AUTHOR]