### Sequences

#### Jumping to Later Terms of GPs

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# Learning Objectives

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###### Jumping to Later Terms of GPs

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**

**Tₙ = a × rⁿ⁻¹**

** **

The rule in this format is known as the **n****th term rule for a GP.**

###### Example

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

Find the 18th term.

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.

T₁₈ = 6929.5 × 1.06¹⁷ = 18659.57.

T₁ = 6929.5.