Dear Pavan
i) Let the first term of the AP be 'c' and its common difference be 'd'.
ii) So its pth term is: c + (p - 1)d = a ------ (1)
and its qth term is: c + (q - 1)d = b ----- (2)
Adding (1) & (2): 2c + (p + q - 2)d = a + b
==> 2c + (p + q - 1)d - d = a + b
==> 2c + (p + q - 1)d = (a + b) + d ---------- (3)
iii) Operating (1) - (2): (p - q)d = (a - b)
==> d = (a - b)/(p - q) --------- (4)
iv) Sum to (p + q) terms is:
S₍p+q₎ = {(p + q)/2}*{2c + (p + q - 1)d}
Substituting from (3) and (4), we get
S₍p+q₎ = {(p + q)/2}*{(a + b) + (a - b)/(p - q)} ---- [Proved]
Regards
Arun (askIITians forum expert)
