Question

Hancox Homes is a popular construction company that builds affordable housing. When the company first started, they sold 1 home the first month, 3 homes the second month, 9 homes the third month, and 27 homes the fourth month. DESCRIBE THE PATTERN THEN REPRESENT THE SEQUENCE AS A NUMERIC SEQUENCE AND A TABLE OF VALUE

Answer 1

The pattern is given below

1, 3, 9, 27

We can clearly see that the terms are indeed the powers of 3

i.e

$\begin{gathered} 3^0\text{ = 1} \\ 3^1\text{ = 3} \\ 3^2\text{ = 9} \\ 3^3\text{ =27} \end{gathered}$

looking closely at the sequence generated, we can show that it follows the geometric sequence as

$r\text{ = }\frac{T_2}{T_1}\text{ = }\frac{T_3}{T_2}\text{ = }\frac{T_{n+1}}{T_n}$

Thus, the common ratio r equals

$r\text{ = }\frac{3}{1}\text{ = }\frac{9}{3}\text{ = }\frac{27}{9}\text{ = 3}$

for any two consecutive numbers in the sequence

Hence, we can express the sequence as a numeric sequence thus:

$\begin{gathered} \text{Recall that T}_n=ar^{n-1} \\ T_n=(1)(3)^{n-1} \\ T_n=3^{n-1} \end{gathered}$

We can also create a table of values to show that this sequence follows the geometric sequence