# How do you find the next two terms of the geometric sequence 405, 135, 45,...?

Nov 4, 2016

The GP is $405 , 135 , 45 , 15 , 5$

#### Explanation:

GP: " "405, 135, 45, ?. ?

You need to know what value the first term has been multiplied by to give the second term.
Is the same value multiplied by ${T}_{2}$ to give ${T}_{3}$?

This value is called the common ratio ($r$).

${T}_{1} \times r \rightarrow {T}_{2} \text{ and } {T}_{2} \times r \rightarrow {T}_{3}$

$r = {T}_{2} / {T}_{1} \text{ and } r = {T}_{3} / {T}_{2}$

$r = \frac{135}{405} = \frac{1}{3} \text{ and } r = \frac{45}{135} = \frac{1}{3}$

[Note: $\times \frac{1}{3}$ is the same as $\div 3$, but we always multiply in a G.P.]

${T}_{4} = 45 \times \frac{1}{3} = 15$

${T}_{5} = 15 \times \frac{1}{3} = 5$

The GP is $405 , 135 , 45 , 15 , 5$