Use of JR pass for Nozomi

If i'm right the JR_PASS is not valid for
NOZUMI train. Have I pay an extra in case
i would like to use it?
I have to ride from Tokyo to Beppu and from there to Nara. Nozumi seems to be the better
one.

The Japan Rail Pass is not valid at all on nozomi trains. You would have to pay the entire fare.

Using the Nozomi takes about 6 hours 15 minutes.

Using the Hikari takes about 7 hours 25 minutes.

The trains inside are the same and just as comfortable, and in the general scheme of things, both take most of the day. If you caught the Hikari one hour earlier, you would arrive about the same time.

It's hardly worth spending an extra 23,000yen+ for a trip that only takes less than 20% longer.

Those figures were for Tokyo - Beppu

I have been complaining about not being able to ride Nozomi w/ JRPass which non reserved seats are the same fare as Hikari and the reserved are just a little more than Hikari. Japan Rail management doesn't have their heads together as this is their best marketing tool for foreign visitors. Nozomi is quieter ans smoother. Sometimes Nozomi cars are used for Hikari. Please write to JNTO near your home.
Having said that, the travel time difference is not much between Nozomi & Hikari but because of required transfer from one Hikari to another at Shin-Osaka, it takes a lot longer. I share your inconvenience to go from Tokyo/Nagoya/Kyoto to Hiroshima or south.

Acctualy the question is: JR pass YES or NO
I have to make 2 large disst. Tokyo-Beppu,
Beppu-Nara, and several short rides around
Tokyo, Nara-Kyoto etc. In two weeks.
The cheapest JR is about 500$. Hiperdia gives me almost the same .
Am I right?

about
Having said that, the travel time difference is not much between Nozomi & Hikari but because of required transfer from one Hikari to another at Shin-Osaka, it takes a lot longer
This is not entirely true, some Hikari's go from Tokyo to Okayama. Switching trains at Okayama is less inconvenient because you don't have to go from one platform to another and my personal experience is, that it's easier to find a free seat when switching trains at Okayama (or Himeji)

A straight round trip ticket between Tokyo and Beppu without reserved seats runs about Y45,000, the same as the 14 day Ordinary pass. The 7-day JR Pass is Y28,300. If you want reserved seats you the RT price is more. If you break your trip, stopping in Kyoto, etc. the price will be more.

The 7-day saves you a lot and the 14 day would save you some and be convenient as you wouldn't have to buy any tickets for local rides on JR. You would have to schedule the Hikari Shinkansen from Tokyo to Hakata with the JR Pass as the Nozomi is set up for xenophobes and JR Pass holders (foreigners) can't use it.