How much money would you have in 15 days if you get a penny doubled each day?

For the fifteen days,

1, 2, 4, 8, 16, 32,

Doubling continues until after the 15th day. But since counting started after 1 penny is given, so n for the sequence would have to be = 16.

This sequence is a GP   common ratio = 4/2 = 2/1 = 2

Sum of nth term of GP =    a(r^n - 1) / (r -1)

where a = 1st term = 1,

n = 16, r = 2

Sum        =        1*(2^16  - 1) /( 2 -1)

                       =  (2^16  - 1) /( 1)

                       =  (2^16  - 1)   =  65536 - 1 = 65535

Money in 15 days = 65 536 pennies.