Kurssi päättyy ensi viikolla.

Breakdown of Kurssi päättyy ensi viikolla.

kurssi
the course
päättyä
to end
ensi viikolla
next week
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Questions & Answers about Kurssi päättyy ensi viikolla.

Why is there no word for the before kurssi?
Finnish does not have articles like a, an or the. Definiteness or indefiniteness is understood from context rather than marked by a separate word.
What grammatical case is kurssi in, and why?
kurssi is in the nominative singular—the default form for the subject of a sentence in Finnish.
What does päättyy mean, and what form is it?

päättyy is the 3rd person singular present of the intransitive verb päättyä, meaning “to end”.
päättyä (infinitive “to end”) → päättyy (“it ends”)

Why is ensi viikolla in the adessive case (-lla) instead of using a preposition like in English?

Finnish expresses many notions that English handles with prepositions by using cases. The adessive case (-lla/-llä) is commonly used for time expressions to mean “during/at (that period)”.
viikko (“week”) → viikolla (“in/at/during the week”)
So ensi viikolla = “next week.”

Why is ensi unchanged (not declined) even though viikolla has a case ending?
ensi is an indeclinable time word in Finnish. Only the noun that follows (here viikko) takes the case ending. If you used a regular adjective instead—like seuraava (“following”)—you would decline it: seuraavalla viikolla.
Could I say Ensi viikolla kurssi päättyy instead? Does word order matter?

Yes. Finnish has relatively free word order. Placing ensi viikolla at the start simply emphasizes the time:
Ensi viikolla kurssi päättyy.
Kurssi päättyy ensi viikolla.

Can I use a different verb instead of päättyä to express “ends”?

Yes. A common synonym is loppua (“to finish/end”). You could say:
Kurssi loppuu ensi viikolla.
The nuance is almost the same in this context.

How do I pronounce päättyy, especially the double letters?

In IPA it’s [ˈpæːtːyː].
ää is a long front vowel (like “ae” in “cat” but lengthened).
tt is a geminate (long) consonant—you hold the t a bit longer.
yy is a long y sound (like the German ü).
Stress falls on the first syllable: PÄÄ-ttyy.