I notice that in the sentence, "there are four bananas" you say "banana wa yon hon DESU" instead of "banana wa yon hon ARIMASU." Is the difference simply that desu is less formal? This caught my eye because I'm used to thinking that arimasu is used to mean "exist", as in sentences that say "there is/are." Thanks.
You have to remember what DESU does. DESU has a wide range of meanings including, is, am, are etc. You are right that ARIMASU can also be used. There are many cases where DESU and ARIMASU have the same function in the sentence:
It is over there.
Asoko desu. OR Asoko ni arimasu.
How many pencils are there?
Enpitsu wa nanbon desu ka. OR Enpitsu wa nanbon arimasu ka.
Where is the car?
Kuruma wa doko desu ka. OR Kuruma wa doko ni arimasu ka.
There are times when ONLY ARIMASU can be used:
Do you have a pen?
Pen ga arimasu ka?
There is a mirror on top of the TV.
Terebi no ue ni kagami ga arimasu.
If you feel more comfortable saying ARIMASU then you should use it. After Lesson 1 in Course 2 the lessons will use ARIMASU quite often. But in Course 1 since we haven't taught ARIMASU yet we are limited to using DESU.
a trick.. Tenchi here is my trick for the "arimas"
I would say that instead of thinking of it like that, think of it as "desu" ="is", so "terebi no ue ni kagami ga arimas" (as shown in example above) means "THERE IS a mirror on the t.v" NOT " IT IS A MIRROR on top of the t.v" Arimas to me is easier to remember as a "Do you have" or "is there..there is/there are"