Sums of Geometric Sequences

Example on evaluating sums of geometric sequences.

Example 1


What is the next term in the geometric sequence $1$,$-\dfrac{1}{3}$,$\dfrac{1}{9}$,$-\dfrac{1}{27}$,…?

  1. $-\dfrac{1}{81}$
  2. $\dfrac{1}{81}$
  3. $\dfrac{1}{54}$
  4. $-\dfrac{1}{54}$
  5. $-1$

Solution


  1. The sequence given is of the form: $a_1$, $a_2$, $a_3$, ... $a_n$. We first need to calculate the common ratio $r$ given by the division of any two successive terms ($r=\dfrac{a_{n+1}}{a_n}$)in the sequence
    $r=\dfrac{a_{2}}{a_1}$ , where $a_2$ and $a_1$ are the second and first terms in the sequence

    $r=\dfrac{-\dfrac{1}{3}}{1}=-\dfrac{1}{3}$

  2. We now calculate the fifth term using the definition of geometric sequence $a_n=r \cdot a_{n-1} $

  3. We need to calculate $a_5$

    $a_5=a_4 \cdot \dfrac{-1}{3}=\dfrac{1}{81}$

    Answer B