Geometric Sequences & Series (DP IB Analysis & Approaches (AA)): Revision Note

Geometric Sequences

What is a geometric sequence?

• A geometric sequence is a sequence where you multiply by a fixed number each time to get the next term in the sequence

  • The fixed number that you multiply by is called the common ratio,

    • e.g. for 2, 6, 18, 54, 162, …

• A geometric sequence can be

  • increasing if

  • decreasing if

  • alternating if

    • changing between positive and negative

How do I find a term in a geometric sequence?

• The n^th term formula for a geometric sequence is

• Where is the first term and is the common ratio

Geometric Series

How do I find the sum of a geometric sequence?

• The formulae for the sum of the first n terms of a geometric sequence are:

• where

  • is the first term

  • is the common ratio

Sum to Infinity

What is the sum to infinity of a geometric series?

• The sum to infinity of a geometric series, , is the sum of all the terms in a geometric sequence (infinitely many terms)

• For or (written )

  • the sum to infinity is infinity,

  • e.g.

    • This geometric series is said to diverge

• For (written )

  • the sum to infinity is a finite value (called a limit)

  • e.g.

    • The limit is 2

    • This geometric series is said to converge

What is the condition for convergence?

• For the sum to infinity of a geometric series to converge to a finite value (a limit), the condition required is

• where means

 How do I calculate the sum to infinity?

• If , then the sum to infinity of a convergent geometric series can be calculated by the formula

• where

  • is the first term

  • is the common ratio

  • 

• The value calculated by the formula is the limit of the series