We would like to apologize for all of the construction noise that has taken place over the last {amount of time}. It was an interruption of your sleep, concentration and working hours.
We would like to share with you what that construction was working towards: {brief description of what was built/fixed.} By telling you the reason for this inconvenience, we hope you will look more charitably on the disruption of your daily lives.
Thank you for your patience and understanding. We apologize again for any problems this caused you.
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Index of letter of apology templates