pth term of an A P is 'a' and qth term is 'b' prove that sum of (p+q)th - Maths - Sequences and Series

Question

pth term of an A P is 'a' and qth term is 'b' prove that sum of
(p+q)th term is p+q/2[a+b+(a-b/p-q)] plz ans needed soon

V.harshith · Answer

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]

pth term of an A.P. is 'a' and qth term is 'b'. prove that sum of (p+q)th term is p+q/2[a+b+(a-b/p-q)]. plz ans needed soon.....