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Find the nth term of the following series.

i. 3 + 7 + 13 + 21 + .......

ii. 5 + 7 + 13 + 31 + 85 + ..........

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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

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