Learn Finnish: Ajoimme kaupunkiin eksymättä ja pysähtymättä. - We drove to the city without getting lost and without stopping.

They use the third infinitive abessive form, which means "without doing" the action. You make it by taking the strong verb stem and adding the ending -matta or -mättä. Here, eksyä (to get lost) becomes eksymättä and pysähtyä (to stop) becomes pysähtymättä.

Why do both of these words end in -mättä instead of -matta?

This is because of vowel harmony. Since the roots of eksyä and pysähtyä contain front vowels (e, y, ä), their endings must also use the front vowel ä. If the verbs had back vowels like a, o, or u (like nukkua, to sleep), they would take the -matta ending instead (like nukkumatta).

Is ajoimme the past tense of ajaa?

Yes! Ajoimme means "we drove" in the past tense (imperfect). To form this, the final a in the stem of ajaa changes to o before adding the past tense marker -i- and the personal ending -mme. If you used the present tense ajamme, it would mean "we drive" or "we are driving".

Why does kaupunki (city) change to kaupunkiin here?