Question

What's the explicit rule for the sequence 3, -6, 12, -24, 48, ...?

Answer 1

`a_(n)` = 3
`(-2)^(n-1)`

Explanation:

there is a common ratio between consecutive terms , that is

- 6 ÷ 3 = 12 ÷ - 6 = - 24 ÷ 12 = 48 ÷ - 24 = - 2

this indicates the sequence is arithmetic with explicit rule

`a_(n)` = a₁
`(r)^(n-1)`

where a₁ is the first term and r the common ratio

here a₁ = 3 and r = - 2 , then

`a_(n)` = 3
`(-2)^(n-1)`

Answer 2

`\sf (-3)/(2)(-2)^n`

Explanation:

Explicit formulas are used to represent all the terms of the geometric sequence with a single formula.

`\sf \boxed{\bf t_n = ar^(n-1)}`

a is the first term.

r is the common ratio.

r = second term ÷ first term.

3 , - 6 , 12, - 24, 48 ,........

a = 3

r = -6 ÷ 3 = -2

`\sf t_n = 3*(-2)^((n-1))\\\\`

`\sf = 3*(-2)^(n)*(-2)^(-1)\\\\ =3*(-2)^n*(-1)/(2)\\\\= (-3)/(2)(-2)^n`

Check:

`\sf t_1 =(-3)/(2)*(-2)^1=(-3)/(2)*(-2) = 3\\\\\\t-2 = (-3)/(2)*(-2)^2 = (-3)/(2)*4=-6\\\\\\t_3=(-3)/(2)*(-2)^3=(-3)/(2)*(-8)=12`