# Question #9a098

Feb 15, 2017

Thus at $n = 7 \text{ we have } {a}_{7} = 3 \times {4}^{7} = 49152$

#### Explanation:

They will be looking for you to derive an equation that gives the value of any term

Let the term count be $i$
Let the ith term be ${a}_{i}$
Let the last term be ${a}_{n}$

$\textcolor{b r o w n}{\text{Test for arithmetic sequence:}}$
$12 - 3 = 9$
$48 - 12 = 26$

$9 \ne 12$ so this is not an arithmetic sequence. Thus is could be a geometric one.

$\textcolor{b r o w n}{\text{Test for geometric sequence:}}$

$12 \div 3 = 4$
$48 \div 12 = 4$
$192 \div 48 = 4$

The values are all the same so this is a geometric sequence that involves 4 raise to some power

Try:

$i = 1 \to {a}_{1} = 3 \times {4}^{1} = 12$
$i = 2 \to {a}_{2} = 3 \times {4}^{2} = 3 \times 16 = 48$
$i = 3 \to {a}_{3} = 3 \times {4}^{3} = 3 \times 64 = 192$

This works so we have: for any $n \textcolor{w h i t e}{\text{ }} {a}_{n} = 3 \times {4}^{n}$

Thus at $n = 7 \text{ we have } {a}_{7} = 3 \times {4}^{7} = 49152$