after a recent purchase
What purchase? When? The offer from Apple is a promotion for buying a new iPhone, iPad, iPod touch, Apple TV box, or Mac. The offer had to be redeemed within the first 90 days.
offer for one year
For the offer from Apple, the duration of the offer period was reduced to 3 months per July 1st.
Read here how to redeem: Apple TV+ offer - Apple
Read here about the conditions: Apple TV+ offer - Terms and conditions - Apple
Read here if you have trouble redeeming: If you can't redeem your promotion for an Apple subscription - Apple Support
The offer from Apple has to be redeemed on your new eligible Apple device, in the TV app.
For Macs, users have to additionally sign in to the TV app (menu Account), for the redeem option to appear.
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Other offers through partnering companies may have other durations, other conditions, and other redeem procedures.