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Pth and qth terms of an A.P are a and b respectively, prove that the sum of (p+q) terms of the series is P+q/2 {a+b+a-b/p-q}.

Pth and qth terms of an A.P are a and b respectively, prove that the sum of (p+q) terms of the series is P+q/2 {a+b+a-b/p-q}.
 

Grade:11

1 Answers

Arun
25763 Points
3 years ago
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)

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