Christmas Contest - Team Round - Problem 24

A sequence a_n has a_1=12, a_2=24, and a_3=48.

When Harry sees the first three terms of this sequence, he thinks:

“This is a geometric sequence!”

However, when Larry (who is in a concerning and steadily declining mental state of being) sees the first three terms, he instead thinks:

“Wow! This sequence is the result of adding together the respective terms of an arithmetic sequence with nonzero difference and a geometric sequence with a positive ratio!”

Based on their respective thoughts, Harry and Larry form their own version of the sequence a_n. Find the positive difference between Harry’s value of a_5 and Larry’s value of a_5.