**I. Arithmetic Sequence and Series**

Arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant.

For example, the sequence 2, 5, 8, 11, 14, 17, . . . is an arithmetic sequence with common difference of 3.

In an arithmetic sequence , First term is \(u_{1}\) , Common difference is \(d\) , General term of a sequence \(u_{n}\), Sum of first n terms \(S_{n}\)

General form of an arithmetic sequence is \(u_{ 1 },\quad u_{ 1 }+d,\quad u_{ 1 }+2d,\quad u_{ 1 }+3d,\quad u_{ 1 }+4d,…………….u_{ 1 }+(n-1)d\)

The nth term or general term of arithmetic sequence is \({ u }_{ n }=u_{ 1 }+(n-1)d\)

Sum of the first terms of an arithmetic sequence is \(S_{ n }=\frac { n }{ 2 } [2u_{ 1 }+(n-1)×d]\)

If first term \(u_{1}\)and last term \(u_{n}\) of an arithmetic series is given then the sum if the series is \(S_{ n }=\frac { n }{ 2 } [u_{ 1 }+u_{ n }]\)

That is Sum up to infinity in arithmetic sequence is infinite only, so sum up to infinity is not possible in arithmetic sequence.

**II. Geometric Sequence and Series**

Geometric sequence is a sequence of numbers such that the ratio or division of two consecutive terms is constant. For example, the sequence 2, 6, 12, 24, 48, 96, . . . is a geometric sequence with common ratio of 3.

In a geometric sequence , First term is \(u_{1}\) , Common ratio is \(r\) , General term of a sequence \(u_{n}\), Sum of first n terms \(S_{n}\)

General form of a geometric sequence is \(u_{ 1 },\quad u_{ 1 }×r,\quad u_{ 1 }×r^{ 2 },\quad u_{ 1 }×r^{ 3 },\quad u_{ 1 }×r^{ 4 },……………u_{ 1 }×{ r }^{ n-1 }\)

The nth term or general term of geometric sequence is \({ u }_{ n }\quad =\quad u_{ 1 }×{ r }^{ n-1 }\)

Sum of the first terms of a Geometric sequence is

\(S_{ n }=\frac { u_{ 1 }×(r^{ n }-1) }{ (r-1) } \) if the common ratio | r | > 1

\(S_{ n }=\frac { u_{ 1 }\quad ×\quad (1-r^{ n }) }{ (1-r) } \) if the common ratio | r | < 1

- Sum up to infinity in geometric sequence is possible only if the common ratio | r | < 1

\(S_{ \infty }=\frac { u_{ 1 } }{ (1-r) } \)

**Skill Builder Questions**