Sequence Formulas

About Sequence Formulas

Arithmetic sequence formulae and geometric sequence formulas are the two forms of sequence formulas. The difference between every two successive terms in an arithmetic series is always the same. A geometric sequence is one in which every two consecutive terms have the same ratio.

The formulas for obtaining the nth term and the sum of the first n terms of each arithmetic and geometric sequence are included in the sequence formulas. Here are the sequence formulas, which will be detailed in detail after the formula list.

  • Arithmetic Sequence:
  • a, a + d, a + 2d,...
  • nthterm: Explicit Formula
  • an = a + (n-1)d
  • Imlicit Formula
  • an = a(n-1) + d
  • Sum of n terms:
  • Sn = n2[2a + (n-1)d](or) Sn = n2[a + an]
  • Geometric Sequence:
  • a, ar2,ar3,....
  • nthterm: Explicit Formula
  • an = a.r(n-1)
  • Imlicit Formula
  • an = r. a(n-1)
  • Sum of n terms:
  • Sn = a(1-rn)1-r
  • Sum of infinite series:
  • S∞ =a1-r

Arithmetic Sequence Formulas

Consider arithmetic sequence a, a + d, a + 2d,..., with an as the first term and d as the common difference. The following are the formulas for arithmetic sequence.

  • nth term of arithmetic sequence (explicit formula),an = a + (n - 1) d.
  • nthterm of arithmetic sequence (implicit formula), an = an −1 + d.
  • The sum of n terms of an arithmetic sequence is, Sn = (n/2) [ 2a + (n - 1) d ] (OR) Sn = (n/2) [a + an] Here,an is the last term of the sequence.

The nth term of an arithmetic series can be found using both explicit and implicit formulas. When values of 'a, n, and d' are known, we use the explicit formula, and when we just know the preceding term and the common difference, we use the implicit formula.

Geometric Sequence Formulas

Consider the geometric sequence a, ar, ar2,... in where an is the first term and d is a common difference. The formulas for the geometric sequence are listed below.

  • nth term of the geometric sequence (explicit formula) is, an = a · rn - 1.
  • nth term of the geometric sequence (implicit formula) is, an = an−1· r.
  • Sum of the n terms of geometric sequence is, Sn = a (rn - 1) / (r - 1).
  • The sum of infinite terms of geometric sequence (when |r| < 1) is, Sn = a / (1 - r).

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Sequence Formulas
Sequence Formulas

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