Question

Thirty-one books are arranged from left to right in order of increasing prices. The price of each book differs by $2 from that of each adjacent book. The price of the book at the extreme right is equal to the price of the middle book plus a book adjacent to the middle one. Give an expression for the cost of the $n$th book in terms of $n$.

Answer 1

Let the middle book cost x dollars. The book to the left costs x-2 and the one to the right costs x+2. The last book, 15 to right of the middle, costs x + 30 dollars. Let's consider two cases:
Case 1: The last book costs the same as the middle book and one to its right. Then
x + (x+2) = x + 30 
x = 28

However, this means the leftmost book costs x-30 = -2 dollars, which is unlikely.
Case 2: The last book costs the same as the middle book and one to its left. Then
x + (x-2) = x + 30
x = 32

In this case, the costs of the books are
$2, $4, $62.
The nth book costs 2n dollars.