(i) ( 1, 4, 9, \ldots )

These are perfect squares: ( 1^2, 2^2, 3^2, \ldots )

( n^{th} ) term: ( a_n = n^2 )

(ii) ( 1, 1+2, 1+2+3, \ldots )

Partial sums of natural numbers (triangular numbers).

( n^{th} ) term: ( a_n = \dfrac{n(n+1)}{2} )

(iii) ( a_1 b_1, a_2 b_2, a_3 b_3, \ldots )

( n^{th} ) term: ( a_n b_n )

(iv) ( x, 2x^2, 3x^3, \ldots )

( n^{th} ) term: ( a_n = n x^n )

(v) ( a_1, a_1 + d, a_1 + 2d, \ldots )

This is an arithmetic sequence.

( n^{th} ) term: ( a_n = a_1 + (n-1)d )

(vi) ( a_1, a_1 r, a_1 r^2, \ldots )

This is a geometric sequence.

( n^{th} ) term: ( a_n = a_1 r^{n-1} )

(vii) ( \dfrac{a_1}{b_1 + c_1}, \dfrac{2a_2}{b_2 + c_2}, \dfrac{3a_3}{b_3 + c_3}, \ldots )

( n^{th} ) term: ( a_n = \dfrac{n a_n}{b_n + c_n} )
(Note: the subscript ( n ) on the right-hand side uses the given sequences ( a_n, b_n, c_n ).)

(viii) ( \dfrac{1}{a_1}, \dfrac{1}{a_1 + d}, \dfrac{1}{a_1 + 2d}, \ldots )

( n^{th} ) term: ( a_n = \dfrac{1}{a_1 + (n-1)d} )