For the following geometric sequence, determine a1 and r and find an explicit formula for the sequence

Question

1)   2, -6, 18, -54…

a)   What is a₁?

b)   What is the common ratio r?

c) Write an explicit formula for a_n using the formula for geometric sequences a_n = a₁*(r)^{n – 1}.

Answer 1

a) The value of a₁ = 2 (the first element of the sequence)

b) To go from 2 to -6 you multiply 2*-3 = -6

To go from -6 to 18 you multiply -6*-3 = 18

etc.

We conclude the common ratio, r, is -3: r = -3

c) a₁ = 2

a₂ = -6 = 2*(-3)

a₃ = 18 = 2*(-3)*(-3)

a₄ = -54 = 2*(-3)³

...

The pattern is:

a_n = 2(-3^n-1) ; n = 1,2,3, ...