Question

What is the formula for 1, -3, 9, -27?
arithmetic or geometric?
​

Answer 1

The formula of the geometric sequence is a
`_(n)` =
`(-3)^(n-1)`

Explanation:

The terms of the sequence are 1, -3, 9, -27

∵ -3 ÷ 1 = -3

∵ 9 ÷ -3 = -3

∵ -27 ÷ 9 = -3

∴ There is a common ratio between each two consecutive terms

∴ The sequence is geometric

∵ The formula of the geometric sequence is a
`_(n)` = a
`r^(n-1)`, where

• a is the first term

• r is the common ratio

∵ The first term is 1

∴ a = 1

∵ The common ratio is -3

∴ r = -3

→ Substitute them in the formula above

∵ a
`_(n)` = (1)
`(-3)^(n-1)`

∴ a
`_(n)` =
`(-3)^(n-1)`

∴ The formula of the geometric sequence is a
`_(n)` =
`(-3)^(n-1)`