He played basketball alone in the park to chase after new friendship.
He played basketball alone in the park to chase for new friendship.
None of them sound particularly correct, but I can't think of a better way to phrase this.
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There are a few problems with this, in my opinion.
Firstly, the phrasal verb
chase after is usually used in literal situations where someone is actually pursuing someone else.
I chased after the pickpocket who stole my wallet.
The dog chased after the cat but couldn't catch it.
If you want to use 'chase' figuratively, you can use
chase + Noun. For example:
We are always chasing happiness but we never seem to achieve it.
She spent her life chasing fame and fortune.
That guy only cares about chasing women. (approaching women for romantic purposes)
This usage often has a negative connotation. The implication is that the 'chasing' is too hurried or hasty, or that the process involved might be questionable.
Finally, 'chasing friendship' sounds a little unnatural because, as mentioned,
chasing + Noun tends to be used to question the validity of the pursuit. There is nothing questionable about wanting to make friends, so it sounds a little strange (to me). I would use a different phrase, such as:
He played basketball alone in the park, hoping to find friendship.
He played basketball alone in the park, hoping to make new friends.