i. 3 + 7 + 13 + 21 + .......
1st consecutive differences 4, 6, 8,........are in A.P.
∴ Tn = an2 + bn + c
Putting n = 1,2,3,
a + b + c = 3, 4a + 2b + c = 7, 9a + 3b + c = 13
⇒ a = 1, b = 1, c = 1
∴ Tn = n2 + n + 1
ii. 5 + 7 + 13 + 31 + 85 +..........
1st consecutive differences 2, 6, 18, 54,....... are in G.P. with common ratio 3
∴ Tn = a.3n–1 + bn + c
Putting n = 1,2,3 we get
a + b + c = 5, 3a + 2b + c = 7, 9a + 3b + c = 13
⇒ a = 1, b = 0, c = 4
∴ Tn = 3n–1 + 4