Question

How do you find the 7th term in the geometric sequence 2, 6, 18, 54, ...?

Answer 1

`4374`

Explanation:

In a geometric sequence, you choose a first term `a`, a ratio `r`, and then you obtain every term multiplying the previous one by the ratio. Let's compute some terms:

• `a_0 = a`
• `a_1 = a*r`
• `a_2 = (a*r)r = a*r^2`
• `a_3 = (a*r^2)r = a*r^3`

and so on. We can see that the relation is `a_n = a*r^n`, which means that the seventh term is `a_7=a*r^7`.

In your case, the first term `a` is `2`, and the ratio can be easily computed: if `a_0=2` and `a_1=6`, then `a_1=a_0*r=6`, which means `2*r=6`, and finally `r=3`.

So, the seventh term will be `2*3^7 = 2*2187 = 4374`