The reason persons with this syndrome age faster is because they have a mutation on the LMNA gene, which causes the production of a permanently farnesylated mutant lamin A called progerin, and this contributes to premature aging. On normal proteins, this farnesyl group would be removed, but people with progeria are not able to remove it and therefore they have alterations.
Rare syndromes part 1: The hutchinson-Gilford progeria syndrome

I've just looked for some images, and i'm really sad for the people who have it. The have a much more limited lifetime than us
That syndrome is very rare. According to cleveland clinic statistics, it happens approximately 1 time per every 4 million births worldwide.
There is a case of a woman with this syndrome that managed to be still alive, she's a bit older than 40 years if im correct
The purpose of this thread is merely informational.

Did you know that there is a genetic disorder called the progeria syndrome? It's a syndrome where the people who suffer from it age much faster than a normal person, estimated to be between 5 to 10 times faster.
In average, persons who suffer this syndrome live between 8 to 21 years, and they usually pass out because of hearth diseases that normally appear in elder people.
Here's an image of how people with this syndrome can look like:
This image corresponds to a popular person with this syndrome called Sam berns. He passed away at the age of 17 (rest in peace).