Sequences

Jumping to Later Terms of GPs

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Jumping to Later Terms of GPs
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If we apply the same thinking we used with basic sequences, we can setermine that geometric sequences follow the below format:

 

a, ar, ar², ar³, ar⁴, ar⁵, where:

 

  • a = the first term; and

  • r = the multiplying rate.

 

This allows us to determine that geometric progressions have an nth term given by the rule:

 

Tₙ = a × rⁿ⁻¹

 

The rule in this format is known as the nth term rule for a GP.


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Example
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The 15th term of a GP Is 15,667 and the 21st term is 22,224.


  1. Find the 18th term.

  2. Find the first term.

   

Solution:

 

To do this we need to first construct a pair of simultaneous equations using the nth term rule.


15667 = a × r¹⁴


22224 = a × r²⁰


Solve the simultaneous equations to find the value of r.


r = 1.06


Substitute the value of r into one of the equations to find the value of a.


a = 6929.5.


We can now answer the two questions.


  1.  T₁₈ = 6929.5 × 1.06¹⁷ = 18659.57.

 

  1. T₁ = 6929.5.

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