Let a 1 , a 2 , ........a n , be real numbers such that
= 
(a 1 + a 2 + ......+ a n ) – 
then find the value of 
Text Solution
Verified by Experts5050
(5050)
Sol. Let 
= b 1 ; 
= b 2 ; 
= b 3 ; ................... 
= b n
∴ b 1 + b 2 + ...... + b n = 
[b 12 + (b 22 + 1) + (b 32 + 2) + ...... + (b n2 + (n – 1))] – 
∴ Σ b 1 = 
[(b 12 + b 22 + b 32 + .......+ b n2 ) + (1 + 2 + 3 + .......+ (n – 1)] – 
⇒ 2 Σ b 1 = Σ b 12 + 
– 
⇒ 2 Σ b 1 = Σ b 12 + n
∴ Σ b 12 – 2 Σ b 1 + Σ 1 = 0 ⇒ 
b 1 – 1 = 0 ⇒ b 12 = a 1 = 1
b 2 – 1 = 0 ⇒ b 22 = a 2 – 1 = 1 ⇒ a 2 = 2
b 3 – 1 = 0 ⇒ b 32 = a 3 – 2 = 1 ⇒ a 2 = 3 and soon
hence a n = n ∴ 
= 1 + 2 + 3 + .... + 100 = 5050
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