Terahertz applications require high performance and high reliability terahertz radiation sources, especially the urgent demands of high output power and broad bandwidth. The extended interaction klystron (EIK) has the great potential to generate hundreds of watt output power in terahertz band. The terahertz EIK adopts multiple gap cavities and unequal-width slots structure is proposed with methodological improvement of bandwidth and output power. The unequal-width slots are the key design of the multiple gap cavity, and the influences of unequal-width slots on the electromagnetic field distribution and beam-wave interaction are analyzed in detail. With multiple gap cavities and unequal-width slots structure, EIK has advantages of wider frequency separation and larger effective characteristic impedance. Particle in cell (PIC) simulation indicates that the bandwidth of unequal-width slots structure can reach to 550 MHz in our initial G-band EIK design. Then, we utilize two kinds of resonance cavities with different width ratios to build a six-cavity beam-wave interaction system and make it operate at the state of stagger-tuning, the bandwidth can be extended to 1-1.5 GHz. Our research shows that the unequal-width slots structure has wider tuning frequency range. Furthermore, the bandwidth can be further broadened to over 2 GHz when dynamic-tuning is adopted, while maintains a high output power of 560 W with efficiency of 11.3% and gain of 47.5 dB. Thus, the methods of multiple gap cavities with unequal-width slots structure, stagger-tuning and dynamic-tuning are much important for the bandwidth improvement of EIK in terahertz band.