The theory that you are familiar with is a little off. What stars can produce is solely a function of size, not generation. Already fused material from a previous star does not allow the new star to fuse more elements. Likewise, the longevity of stars is solely a function of size. It’s a balance between the heat of fusion and the pressure of gravity. More matter in the star means more pressure, which means the rate of fusion increases and more elements can be fused, but the fuel is consumed significantly faster.
The smaller a star is the longer it burns, because there is less pressure being exerted by gravity to drive the fusion process. Big stars don’t last long (the biggest only a few million years), but they produce the all of the naturally occurring elements—up to iron via normal fusion, and the heavier elements during supernova that occurs after iron fusion begins. Smaller stars like our sun will never get past the carbon stage and will never go supernova, and smaller stars still like brown dwarfs will never get past the hydrogen stage. These small stars last the longest because their rate of fusion is incredibly slow.
The theory that you are familiar with is a little off. What stars can produce is solely a function of size, not generation. Already fused material from a previous star does not allow the new star to fuse more elements. Likewise, the longevity of stars is solely a function of size. It’s a balance between the heat of fusion and the pressure of gravity. More matter in the star means more pressure, which means the rate of fusion increases and more elements can be fused, but the fuel is consumed significantly faster.
The smaller a star is the longer it burns, because there is less pressure being exerted by gravity to drive the fusion process. Big stars don’t last long (the biggest only a few million years), but they produce the all of the naturally occurring elements—up to iron via normal fusion, and the heavier elements during supernova that occurs after iron fusion begins. Smaller stars like our sun will never get past the carbon stage and will never go supernova, and smaller stars still like brown dwarfs will never get past the hydrogen stage. These small stars last the longest because their rate of fusion is incredibly slow.