All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
Our answer to your question will be summarized as follows:
Firstly, if you want to make up missed fasts days by fasting consecutively, then one intention is sufficient on the first day according to the view of some scholars. Therefore, your fast is correct. However, to be on the safe side, it is bed act according to the opinion of the majority of the scholars who stated that it is a condition for the validity of fasting for one to renew the intention every day because it is not an obligation to fast make-up days consecutively.
Secondly, the predominant opinion regarding the correct time of making the intention to fast according to the majority of the scholars, is that one must make the intention the night before (i.e., before Fajr); it is a condition for the validity of the fast. This is the case whether one is fasting a due obligatory fasting or making up for a missed obligatory fast. This because the Prophet said: “Whoever does not make his intention to fast before Fajr, his fasting will not be accepted.” [Ahmad, Abu Daawood, At-Tirmithi, An-Nasaa’i, and Ibn Maajah] [Al-Albaani classified it as authentic]
Thirdly, we draw your attention to the fact that the intention is in the heart and it is not uttered by the tongue. So, it is not a condition for you to say “I will fast” or “I intend to fast”. As a matter of fact, it is not permissible to utter the intention.
Allah knows best.