Sequences
Jumping to Later Terms of GPs
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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
The rule in this format is known as the nth 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.