Question

Find the first five terms of the sequence in which a1 = 6 and an = –3an – 1 – 12, if n ≥ 2.

Answer 1

`\huge\boxed{6,\ -30;\ 78;\ -246;\ 690}`

Explanation:

`a_1=6\\a_n=-3a_(n-1)-12\qquad n\geq2\\\\\text{Substitute}\ n=2,\ n=3,\ n=4,\ n=5\ \text{to}\ a_n=-3a_(n-1)-12:\\\\\text{for}\ n=2\\\\a_2=-3a_(2-1)-12=-3a_1-12\\\\\text{substitute}\ a_1=6\\\\a_2=-3(6)-12=-18-12=-30\\\\\text{for}\ n=3\\\\a_3=-3a_(3-1)-12=-3a_2-12\\\\\text{substitute}\ a_2=-30\\\\a_3=-3(-30)-12=90-12=78`

`\text{for}\ n=4\\\\a_4=-3a_(4-1)-12=-3a_3-12\\\\\text{substitute}\ a_3=78\\\\a_4=-3(78)-12=-234-12=-246\\\\\text{for}\ n=5\\\\a_5=-3a_(5-1)-12=-3a_4-12\\\\\text{substitute}\ a_4=-246\\\\a_5=-3(-246)-12=702-12=690`