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A sequence is defined by the explicit formula [tex]a_n=3^n+4 [/tex]. Which recursive formula represents the same sequence of numbers?

A. [tex]a_n=3a_n_-_1+4;a_1=7 [/tex]

B. [tex]a_n=3a_n_-_1-8;a_1=7[/tex]

C. [tex]a_n=3a_n_-_1-8;a1=7[/tex]

D. [tex]a_n=n^3+6;a_1=7[/tex]

Respuesta :

Let's write explicitly two consecutive terms:


[tex] a_n = 3^n+4,\qquad a_{n-1} = 3^{n-1} + 4 = \frac{3^n}{3} + 4 [/tex]


This means that


[tex] 3a_{n-1} = 3^n + 12 [/tex]


which is 8 more than we expected. So,


[tex] 3a_{n-1}-8 = 3^n + 12-8 =3^n + 4 = a_n [/tex]


This is options B and C, since their statements seem the same

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