Why does the astable mode in 555 timer require two resistors, whereas monostable works with just one resistor ? The cap gets charged through R1 and R2 and gets discharged through R2. But in monostable it gets discharged without any resistor.
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Resistor is not a essential component in capacitor discharge path. It can be used to limit the current flow thereby increasing discharge time of the capacitor.
Charging and discharging time of capacitor determines time period of two states of a astable 555 timer. In your circuit R2 determines discharge time.
In mono_stable 555 timer, astable state's time duration is determined by the charging time of the capacitor. Second stage is the stable state therefore discharge time of the capacitor is not critical here.