Mathematics

Problem

In an arithmetic progression, the sum of the first

n

terms is one–sixth of the sum of the first

3n

terms, where

n

is a positive integer (

n > 1

). a. (i) Establish an algebraic relation connecting

n

, the first term

a

, and the common difference

d

that follows from the given condition. b. (ii) Show that the

n^\text{th}

term is the arithmetic mean of the first term and the

3n^\text{th}

term. c. (iii) If the common difference is

2

and

n = 6

, find the first term and the

18^\text{th}

term.