Question

How do you write the first five terms of the sequence defined recursively `a_1=81, a_(k+1)=1/3a_k`, then how do you write the nth term of the sequence as a function of n?

Answer 1

Given: `a_1 = 81` and `a_(k+1) = 1/3a_k`

Starting with the first term:

`a_1 = 81 = (1/3)^0 81`

`a_2 = (1/3)81 = (1/3)^1 81`

`a_3 = (1/3)a_2 = (1/3)(1/3)81 = (1/3)^2 81`

`a_4 = (1/3)a_3 = 1/3(1/3)^2 81 = (1/3)^3 81`

The equation for the nth term is:

`a_n = (1/3)^(n-1)81`