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Initial Tier 1 General visa date: 29/10/2009
Entry date in the UK: 17/02/2010
Expiry date: 29/10/2012

Extension (BRP) date: 25/01/2013
BRP expiry date: 25/01/2015

Q1) Can I apply for ILR on the 20th of January or is it the 21st?
Q2) Does the gap between my initial tier 1 expiry date and BRP issue date affect my application or do i need to write a letter or anything?

Regards

28 days before 25/01/15.

Nothing should affect ur application,

From 28 days before 17/02/2015.

B1 English and LIUK required.

As you entered after 90 days when initial visa granted, therefore you will count you period from your date of entry. The Qualifying period will be calculated as follows:
Entry date in the UK: 17/02/2010
BRP expiry date: 25/01/2015
Your qualifying period for five years will be completed on 16/02/2015.
28 days before 16/02/2015 will be 19/01/2015
Therefore you can easily apply within your validity date

Thanks