Geometric Sequence Formulas

About Geometric Sequence Formulas

Multiple formulas linked to a geometric sequence are included in the geometric sequence formulas. Let us first define a geometric sequence before understanding these formulas. It's a numerical sequence in which the ratio of every two subsequent numbers is always the same. 2, 4, 8, 16,..., for example, is a geometric sequence because the common ratio of every two consecutive terms is 2, i.e. common ratio = 4/2 = 8/4 = 16/8 =...... = 2.

What do you mean by Geometric Sequence Formulas?

The formulas for obtaining the nth term and the sum of the n terms are included in the geometric sequence formulas. When the common ratio of a geometric series is smaller than 1, we can also get the sum of infinite terms. The geometric sequence formulas for a geometric sequence with its first term 'a' and common ratio 'r' will be discussed (i.e., the geometric sequence is of form a, ar, ar2, ar3, ....).

  • an = a· rn - 1
  • Sn = a (rn- 1) / (r - 1)
  • Sn = a (1 - rn) / (1 - r).
  • S∞ a / (1 - r).

Let's take closer look at each of these formulas.
Geometric Sequence Formula nth Term

We have considered the sequence to be a, ar, ar2, ar3, ....... Its first term is a (or ar1-1), its second term is ar (or ar2-1), its third term is ar2(or ar3-1). Thus,

The geometric sequence's nth term is, an= a · rn - 1.

Sum of Geometric Sequence Formula upto n terms

The sum of the geometric sequence a, ar, ar2, ar3, .......... upto 'n' terms is,

  • Sn = a + ar + ar2+......+arn-1.....(1)
  • When both sides are multiplied by 'r,'
    rSn = a + ar + ar2+......+arn....(2)
  • Equation (2) is subtracted from equation (1),
    Sn - rSn = a - arn
    Sn(1 - r) = a(1 - r)n

Both sides are divided by (1 - r),
We can also derive the other formula(Sn = a (rn- 1) / (r - 1). Thus,

The sum of the geometric sequence's first 'n' terms is Sn = a (1 - rn) / (1 - r), when |r| < 1
[OR]
Sn = a (rn- 1) / (r - 1), when r > 1 (or) when r < -1

Get the list of all Maths formulas used in general calculations. 

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Geometric Sequence Formulas

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