Geometric Series

Question

Given that a_n =(-3)(4)^n-1 for a geometric sequence, determine the common ratio:

Answer 1

Hi Giselle,

To find the common ratio, we need to look at a few points.

plug in n = 1

a₁ = (-3)(4)^1-1

a₁ = (-3)(4)^0.

a₁ = -3(1) = -3

^{Do you remember what a number to the 0 power is? It's 1.}

Plug in n = 2

a₂ = (-3)(4)^2-1

a2 = (-3)(4)¹

a2 = (-3)(4) = -12

We could keep going, n = 3, n = 4...on so on.

to find a common ratio take a₂ divided by a_{1 ,} a₃ divided by a_{2, you should get the same number!}

What do you get?

Let me know if you have more questions.

Best,

Miss Julie

Answer 2

Well, we are given a rule: a_n=(-3)(4)^n-1

We can think of this rule as the nth entry of the geometric sequence.

So when n=1,(Substitute n=1 into the rule), a_1=(-3)(4)^1-1=-3(1)=-3

If n=2,a₂=(-3)(4)^2-1=(-3)(4)=-12

if n=3,a₃=(-3)(4)^3-1=-48

Now we have the sequence a₁,a₂,a₃ or -3,-12,-48

To determine the common ratio, we have to ask ourselves how we get from -3 to -12, or from -12 to -48. Well, -3 multiplied by (4) gets us -12, and -12 multiplied by (4) gets us -48. Therefore our common ratio is 4.