Geometric sequence help

Question

Find a formula for the general term a_n of the sequence assuming the pattern of the first few terms continues.

−12/5, 3/5, −18/125, 21/625, −24/3125, ...

Assume the first term is a₁

a_n =

Answer 1

Initially, it is difficult to see a pattern. However, if we write 3/5 as 15/25, it might be easier to see:

-12/5 , 15/25 , -18/125, 21/625 , ...

Now, we can see a pattern in the numerator and a separate pattern in the denominator. Let's write an expression for the numerator first.

12 , 15 , 18 , 21 , ...

The pattern is that each term is 3 more than the previous term. This is an arithmetic sequence and the equation is 3n + 9. Now, let's do the same thing for the denominator:

-5 , 25 , -125 , 625 , ...

The pattern is that each term is -5 times the previous term. This is a geometric sequence and the equation is (-5)ⁿ

So, the formula for the sequence as a whole is an = (3n + 9) / (-5)ⁿ