4.2.8

Geometric Series

Test yourself

Geometric Series

The sum of the terms in a geometric sequence is called a geometric series.

Illustrative background for SeriesIllustrative background for Series ?? "content

Series

  • The sum of the terms of a sequence is called a series.
  • For the geometric sequence defined by an=5×2na_n = 5\times 2^n, the first four terms are:
    • 5,10,20,405,10,20,40
  • The series is then given by:
    • 5+10+20+405 +10+20+40
  • The sum of this series is 75.
Illustrative background for General formulaIllustrative background for General formula ?? "content

General formula

  • We can write the sum of the first nn terms of a geometric series with common ratio rr as:
    • Sn=a1+ra1+...+rn1a1S_n = a_1 + ra_1+ ...+ r^{n-1}a_1
  • Multiplying both sides of the equation by rr we have:
    • rSn=ra1+r2a1+...+rna1rS_n = ra_1 + r^2a_1+...+ r^na_1
Illustrative background for General formulaIllustrative background for General formula ?? "content

General formula

  • Subtracting this from the original equation for SnS_n, almost all of the terms cancel out:
    • SnrSn=a1rna1S_n - rS_n = a_1 - r^na_1
  • Taking out a factor of SnS_n and dividing by 1r1-r, we have the general equation for the sum of a geometric series:
    • Sn=a1(1rn)(1r),  r1\large S_n = \large a_1 \frac{\large(1-r^n)}{\large(1-r)},\; r\ne 1

Jump to other topics

1Proof

2Algebra & Functions

2.1Powers & Roots

2.2Quadratic Equations

2.3Inequalities

2.4Polynomials

2.5Graphs

2.6Functions

2.7Transformation of Graphs

2.8Partial Fractions (A2 Only)

3Coordinate Geometry

4Sequences & Series

5Trigonometry

6Exponentials & Logarithms

7Differentiation

8Integration

9Numerical Methods

10Vectors

Go student ad image

Unlock your full potential with GoStudent tutoring

  • Affordable 1:1 tutoring from the comfort of your home

  • Tutors are matched to your specific learning needs

  • 30+ school subjects covered

Book a free trial lesson