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I would like to have a sentence to say I will study those new ideas next.

This is what I wrote:

The following new ideas will be carrying on.

or

The following new ideas will be proceed

I was wondering which of them sounds more formal? And, am I using will be correctly?

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Neither of these is idiomatic. Ideas cannot perform actions like carry on and proceed—it is you who do these things.

If you have just been talking about the ideas you are studying now, you might say that the new ideas will follow or come next. But I really think that something like your opening description is the best approach: "I will be studying these new ideas next." It's simple and clean, with no metaphors. As Mies van der Rohe said, "Less is more".

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Proceed sounds more formal to me (AmE), but your phrasing is very awkward.

I would lead off with the "moving forward" phrase, and follow with the ideas.

Proceeding with ideas A and B, we see that C...

Also, "carrying on" has a connotation of something being boring, though I think BrE and AuE may feel differently.

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