How do you find a formula for the nth term of the geometric sequence: 8, 12, 18, 27, ...?

Mar 9, 2016

$n$th term is $8 \cdot {\left(\frac{3}{2}\right)}^{n - 1}$

Explanation:

The formula for $n$th term of geometric sequence is ${a}_{1} {q}^{n - 1}$, where ${a}_{1}$ is the 1st term and $q$ is the quotient of this sequence (check for $n = 1$).

To find quotien $q$ we must divide any term by its previous one - result will be the same (it's geometric sequence definition).

$q = {a}_{k} / {a}_{k - 1} = {a}_{2} / {a}_{1} = \frac{12}{8} = \frac{3}{2}$ for any $k > 1$.

Moreover

$\frac{12}{8} = \frac{18}{12} = \frac{27}{18} = \ldots$

so the sequence is geometric indeed.

...

Fun fact: next term is not an integer.