The first terms of an infinite geometric sequence,

MATHEMATICS · Multi-part · AoPS · Mathematics · Sequences & Series · Sequences & Series

Problem

u_n

, are 2, 6, 18, 54, ... The first terms of a second infinite geometric sequence,

v_n

, are 2, -6, 18, -54, ... The terms of a third sequence,

w_n

, are defined as

w_n = u_n + v_n

. The finite series,

\sum_{k=1}^{225} w_k

, can also be written in the form

\sum_{k=0}^{m} 4r^k

. (1) Write down the first three non-zero terms of

w_n

. (2) Find the value of

r

. (3) Find the value of

m