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

Nov 10, 2015

$4374$

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:

• ${a}_{0} = a$
• ${a}_{1} = a \cdot r$
• ${a}_{2} = \left(a \cdot r\right) r = a \cdot {r}^{2}$
• ${a}_{3} = \left(a \cdot {r}^{2}\right) r = a \cdot {r}^{3}$

and so on. We can see that the relation is ${a}_{n} = a \cdot {r}^{n}$, which means that the seventh term is ${a}_{7} = a \cdot {r}^{7}$.

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 ${a}_{1} = {a}_{0} \cdot r = 6$, which means $2 \cdot r = 6$, and finally $r = 3$.

So, the seventh term will be $2 \cdot {3}^{7} = 2 \cdot 2187 = 4374$