(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)
n2 - an = n2 - n
⇒ an = n
Therefore, a7 = 7.