Find the first six terms of each of the following sequences, starting with n = 1.

Question

Find a formula a_n for the nth term of the geometric sequence whose first term is a₁ = 1, such that (a_{n + 1} / a_n) = 10 for n ≥ 1.

Find a function f(n) that identifies the nth term a_n of the following recursively defined sequence, as a_n = f(n).

a₁ = 2 and a_{n + 1} = 2a_n for n ≥ 1

Answer 1

a_n+1 / a_n = 10

a_n+1 = 10 a_n, geometric

a_n = a₁(10)^n-1, substitute values and answer

a_n+1 = 2 a_n, geometric

a_n = a₁(2)^n-1, substitute values and answer

Answer 2

Given a_n+1 / a_n = 10

If we put n = 1 in the above equation,

a₂ / a₁ = 10

a₂ = 10 * a₁ = 10 * 1 = 10

In the same equation, if we put n = 2,

a₃ / a₂ = 10

a₃ = 10 * a₂ = 10 * 10 = 100

Similarly,

a_n = 10 * 10 * 10 .. (n-1) times * 1 = 10^n-1

It is always a good idea to check again whether the formula holds true for n=1, 2.

The second problem also follows the same pattern. Let me know if you have trouble applying it to second problem.