Question

What is the formula for this geometric sequence?
1, -3, 9, -27,...

Answer 1

Well, the first thing you can see is that you are constantly switching signs, so you have to have a negative in your sequence. Then, if possible, you can try and recognize the pattern in how the numbers are changing. Going from 1 3 9 27 is an increase by multiples of 3. Since you need to start at 1, you also need your first n-term to be 0. Writing this in the form of ar^n like a geometric sequence, you could put (-3)^n-1. This way, the first term will be 1 when n = 1 because (-3)^(1-1) = (-3)^0 = 1. And, if you were to keep checking subsequent numbers for n, you would see the sign changes back and forth and you keep increasing by multiples of 3. A lot of these geometric series are just pattern recognition. Once you start learning what to look for, they're not that bad. So in the end, the answer would be (-3)^(n-1)

Answer 2

the general formula of geometric sequence is
an = a1 * r^(n-1)

known a1 = 1
r = a2/a1 = -3/1 = -3
therefore,
an = a1 * r^(n-1)
an = 1 * (-3)^(n-1) = (-3)^(n-1)

Answer 3

Thanks for your help guys :)