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in Binomial theorem by (52.7k points)
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If (12 - a1) + (22 - a2) + (32 - a3) + … + (n2 - an) = 1/3n(n2 - 1) then find the value of a7.

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(12 + 22 + … + n2) - (a1 + a2 + … + an) = 1/3 n(n2 - 1) ...(1) Replacing n by (n - 1), then

(12 + 22 + … + (n - 1)2) - (a1 + a2 + … + an-1)

= 1/3(n - 1)((n - 1)2 - 1)   .....(2)

Subtracting Eq. (2) from Eq. (1)

n- a= n- n 

 ⇒ an = n

Therefore, a7 = 7.

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