If you are offered a position with a consulting firm (and are happy with the terms of the offer) then feel free to accept it.
You can talk to the recruitment management to let them know your decision, and you will also need to sign, date, and return a copy of the offer letter.
Be sure to make a copy for your records.
If you have successfully negotiated any terms of the offer, then you need to capture the additional or amended terms in writing.
Ideally the firm should provide you with an updated offer letter reflecting the agreed terms of the offer.
However, the recruitment manager may conveniently forget to provide this, in which case you should set out the agreed terms in an email and send it to the recruitment manager.
The firm may try to wriggle out of its obligations later, and so you need to have written evidence of what was agreed.
[For more information on consulting interviews, please download “The HUB’s Guide to Consulting Interviews“.]