Two students start a club the first week of school. each member invites three new members the second week of school. each of these new members then invites three additional members the following week. if the pattern of each new member inviting three new members each week continues, how many members will be in the club after five weeks

I'll just make a manual computation on how 1 member recruits his members until week 5. Then multiply the sum by 2.
                                old      new
1:   1
2:   1 x 3 = 3 ⇒        1         3       *only new members recruit 3 more
3:   3 x 3 = 9 ⇒        4         9 
4:   9 x 3 = 27⇒     13       27
5: 27 x 3 = 81 ⇒    40       81

40 + 81 = 121. Total number of members under 1 founding member.

121 x 2 founding members = 242 total number of members within 5 weeks.