The test book is correct. You need to prove that the under lying sequence whose sum is given by is in A.P. Did you try considering ?
Spoiler:
I'm back with another potential error in the textbook by Chandler/Bostock, Core Course for A Level. Last time it was my mistake, but this time I am more confused:
Page 594, Question 8
The sum of the first n terms of a series is given by Sn = n(3n-4). Show that the terms of the series are in arithmetic progression.
Can this be right?