let the number of required sequences be Tk
Any strictly increasing sequences of postitive inegers a1,a2,...,ak with a1=1 and ak=n will either have n-1 or it won't have n-1 as its penultimate integer.
when it has ak-1 = n-1 in the sequence the number of such sequences i.e; a1,a2,...,ak-1,ak is counted as Tk-1. ( ak = n is fixed so we count the remaining part that ends in ak-1 = n-1 )
when n-1 is not present as the penultimate integer in the sequence, the number of sequnces is still Tk-1. ( as the sequence effectively becomes a1,a2,...,ak-1 where ak-1 = n )
So, Tk = Tk-1 + Tk-1 = 2Tk-1 with initial conditions T2 = 1.
Solving it in closed form gives Tk = 2k-2 .