Consider an arithmetic progression with first term a and common difference d.
a. (i) Explain why the sum of the terms from the p-th to the q-th term (inclusive), where q>p≥1, can be written as Sq−Sp−1, where Sn denotes the sum of the first n terms of the progression.
b. (ii) Using the result of part (i), obtain a simplified expression, in terms of a, d, p and q, for the sum Tp,q=ap+ap+1+⋯+aq.