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

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