Question

Write the explicit formula for the sequence -1,2,-4,8

Answer 1

Given the following sequence:

`\text{ -1, 2, -4, 8}`

Let's determine its explicit formula:

a.) Let's verify if it's an Arithmetic Sequence. Finding out if the sequence has a common difference.

`\text{ 8 - (-4) = 8 + 4 = 12}`
`\text{ -4 - 2 = -6}`
`\text{ 2 - (-1) = 2 + 1 = 3}`

The sequence doesn't have a common difference. Therefore, the sequence is not arithmetic.

b.) Let's verify if it's a Geometric Sequence. Finding out if the sequence has a common ratio.

`\text{ 8 }/\text{ -4 = -2}`
`\text{ -4 }/\text{ 2 = -2}`
`\text{ 2 }/\text{ -1 = -2}`

The sequence has a common ratio of -2. Thus, the sequence is Geometric. Let's now proceed in completing the equation.

The standard Geometric Sequence Formula:

`\text{ }A_n=A_1(r)^{n\text{ - 1}}\text{ }`

Where,

An = the nth term

A₁ = the 1st term

r = the common ratio

n = the number of terms

From the given sequence, we get:

A₁ = -1

r = -2

Completing the explicit formula, we get:

`\text{ }A_n=A_1(r)^{n\text{ - 1}}\text{ }`
`\text{ }A_n=-1(-2)^{n\text{ - 1}}\text{ = -}(-2)^{n\text{ - 1}}`

Therefore, the explicit formula of the given sequence is An = -(-2)ⁿ⁻¹