Let T1,T2,T3,⋯Tn be the terms of a sequence and let (T2−T1)=T |1,(T3−T2)=T |2,⋯(Tn−Tn−1)=T |n−1 . Case-I: If T |1,T |2,⋯T |n−1 are in A.P., then Tn is quadratic in 'n' . If T |1−T |2,T |2−T |3,⋯ are in A.P., then Tn is cubic in n . Case-II: T |1,T |2,⋯T |n−1 are not in A.P., but in G.P., then Tn=arn+b , where r is the common ratio of the G.P. T |1,T |2,T |3,⋯ and a,b∈R . Again, If T |1,T |2,⋯T |n−1 are not in G.P. but T |2−T |1,T |3−T |2,⋯ are in G.P., then Tn is of the form arn+bn+c a,b,c∈R .

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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 $a r^{n} + b n + c a, b, c \in R$ .