Question

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

Answer 1

Answer:

$a_{n}$ = 3 $(-2)^{n-1}$

Step-by-step 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

Answer:

$\sf \dfrac{-3}{2}(-2)^n$

Step-by-step 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

         $\sf t_n = 3*(-2)^{(n-1)}$

             $\sf = 3*(-2)^{n}*(-2)^{-1}\\\\ =3*(-2)^n*\dfrac{-1}{2}\\\\= \dfrac{-3}{2}(-2)^n$

Check:

$\sf t_1 =\dfrac{-3}{2}*(-2)^1=\dfrac{-3}{2}*(-2) = 3\\\\\\t-2 = \dfrac{-3}{2}*(-2)^2 = \dfrac{-3}{2}*4=-6\\\\\\t_3=\dfrac{-3}{2}*(-2)^3=\dfrac{-3}{2}*(-8)=12$