In the Real World
Looking for patterns in sequences of numbers can be something you do because it's fun, or it can be something you do to get better at finding patterns in other contexts.
Being able to recognize and describe patterns is helpful in almost every area of thought imaginable. To support that statement, here's a selection of wildly different examples.
To decipher the encrypted message
WKH IRA OLNHV WR HDW WKH GXFNV
it's useful to recognize that the pattern WKH occurs twice. There aren't many three-letter words that would occur twice in that short of a message.
A thirteen year old kid made a breakthrough in solar power by recognizing an occurrence of the Fibonacci sequence in nature.
To determine whether a poem is a haiku, sonnet, limerick, or something else, you need to recognize what pattern it follows. If you write a good geeky haiku, you can win stuff.