Question

Consider a sequence whose 3rd term is 13 and whose 7th term is 208. assume the sequence is geometric. find the function a(n) that describes the sequence and find the 19th term of the sequence

Answer 1

For geometric sequence, the general formula is:

An = A₀rⁿ⁻¹

So for n=3 and An = 13
13 = A₀r² 
A₀ = 13/r² --> eqn 1

For n=7 and An = 208
208 = A₀r⁶ --> eqn 2

Solving simultaneously,

208 = (13/r²)(r⁶)
208 = 13r⁴
r = ⁴√208/13 = 2

Then, A₀ = 13/2² = 13/4

1. The general formula would then be: An = (13/4)(2)ⁿ⁻¹
2. If n=19,
An = (13/4)(2¹⁸) = 851,968