Let $T_{1}, T_{2}, T_{3}, \dots T_{n}$ be the terms of a sequence and let $(T_{2} - T_{1}) = T^{|}_{1}, (T_{3} - T_{2}) = T^{|}_{2}, \dots (T_{n} - T_{n - 1}) = T^{|}_{n - 1}$ .
Case-I: If $T^{|}_{1}, T^{|}_{2}, \dots T^{|}_{n - 1}$ are in A.P., then $T_{n}$ is quadratic in $' n'$ . If $T^{|}_{1} - T^{|}_{2}, T^{|}_{2} - T^{|}_{3}, \dots$ are in A.P., then $T_{n}$ is cubic in $n$ .

Case-II: $T^{|}_{1}, T^{|}_{2}, \dots T^{|}_{n - 1}$ are not in A.P., but in G.P., then $T_{n} = a r^{n} + b$ , where $r$ is the common ratio of the G.P. $T^{|}_{1}, T^{|}_{2}, T^{|}_{3}, \dots$ and $a, b \in R$ . Again,

If $T^{|}_{1}, T^{|}_{2}, \dots T^{|}_{n - 1}$ are not in G.P. but $T^{|}_{2} - T^{|}_{1}, T^{|}_{3} - T^{|}_{2}, \dots$ are in G.P., then $T_{n}$ is of the form $a r^{n} + b n + c a, b, c \in R$ .

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