Final answer:
The first red giant stage of a one solar mass star ends when it forms a carbon-oxygen core, signaling the cessation of central nuclear energy production and the star's forthcoming death, often accompanied by the shedding of its outer layers to form a planetary nebula.
Step-by-step explanation:
The first red giant stage of a one solar mass star's life usually ends with the formation of a carbon-oxygen core. This formation marks the end of the generation of nuclear energy at the center of the star, signaling that its death is near. After the core has run out of helium to burn, the star enters its final death throes; it may shed its outer layers, causing the ejection of a planetary nebula and exposing its hot inner layers that will eventually become a white dwarf.
During its red giant phase, a star such as our Sun becomes significantly larger and cooler, leaving the main-sequence band and ascending upwards and to the right on the Hertzsprung-Russell (H-R) diagram. This is due to the outer layers expanding while the core contracts. The luminosity increases even though the surface temperature decreases, which eventually leads to the star shedding its outer layers and the core's temperature rising enough to fuse helium into carbon and oxygen through the triple-alpha process.
In conclusion, a star like the Sun will go through several stages before it reaches the end of its life cycle, including the dramatic red giant phase where it burns helium into heavier elements until it can burn no more and transitions into its final state.