Sequences and Series Formulas

About Sequences and Series Formulas

A sequence is the set of ordered elements that follow a series which is the sum of the elements of a sequence. Sequence and series include the formulas for nth term and sum.

What are Sequences and Series Formulas?

Below list includes sequences and series formulas for the A.P, G.P, and H.P. Where the sequence and series formulas include formulas:

  • To find nth term of the sequence
  • To find sum of n terms of series

In A.P, there is a common difference between two consecutive terms. In G.P , there is common ratio between consecutive terms. In H.P, the reciprocals of its terms are in an arithmetic sequence. The figure below shows all the formulas for Sequence and Series

Arithmetic
Sequence formula of nth term an = a + (n-1)d
Series formula for the sum of n terms Sn = n2[2a + (n-1)d]
Geometric
Sequence formula of nth term an = a.r(n-1)
Series formula for the sum of n terms Sn = a(1-rn)1-r
Series formula for sum of infinite terms Sn = a1-r when |r| < 1
Harmonic
Sequence formula of nth term an = 1a+(n-1)d
Series formula for the sum of n terms Sn = 1dln[2a+(2n-1)d2a-d]

Arithmetic Sequence and Series Formulas

Let arithmetic sequence be a, a+d, a+2d, a+3d, a+4d, ...., where ‘a’ is first term and ‘d’ is common difference. Then:

  1. nth term of AP = an = a + (n-1)d
  2. Sum of the AP = Sn = n2[2a + (n-1)d] (or) Sn = n2[a + an]

Geometric Sequence and Series Formulas

Let geometric sequence be a, ar2,ar3,.... where 'a' is the first term and 'r' is the common ratio. Then:

  1. nth term of GP, an = a.r(n-1)
  2. Sum of finite G.P (sum of first 'n' terms), Sn = a(1-rn)1-r
  3. Sum of infinite G.P, Sn = a1-r when |r| < 1. The sum is not defined when |r| ≥ 1.

Harmonic Sequence and Series Formulas

Let harmonic sequence be 1/a, 1/(a+d), 1/(a+2d), 1/(a+3d), 1/(a+4d), ...., where '1/a' is its first term and 'd' is the common difference of the arithmetic sequence a, a + d, a + 2d, .... Then:

  1. nth term of H.P, an = 1a+(n-1)d
  2. Sum of H.P,Sn = 1dln[2a+(2n-1)d2a-d]

Download all the Maths formulas from the HT maths page.

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Sequences and Series Formulas
Sequences and Series Formulas

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