Question

Write down the first five terms of the sequence:

$a_1=4,a_n=3a_{n-1}+2$.

Short answer

First five terms of the sequence are$4,14,44,134,404.$

Step-by-step solution

Given information:

Consider , $a_n=3a_{n-1}+2$

Given first term, $a_1=4$

To find the values of$a_2,a_3,a_{4}and{a}_5.$

To find a2 :

Put $n=2$,

$$\begin{gathered} a_2=3\times a_{2-1}+2 \\ =3\times a_1+2 \\ =3\times 4+2 \\ =12+2 \\ =14 \end{gathered}$$

To find a3 ,

Put $n=3$,

$$\begin{gathered} a_3=3\times a_{3-1}+2 \\ =3\times a_2+2 \\ =3\times 14+2 \\ =42+2 \\ =44 \end{gathered}$$

To find a4 :

Put $n=4,$

$$\begin{gathered} a_4=3\times a_{4-1}+2 \\ =3\times a_3+2 \\ =3\times 44+2 \\ =132+2 \\ =134 \end{gathered}$$

To find a5 :

Put $n=5,$

$$\begin{gathered} a_5=3\times a_{5-1}+2 \\ =3\times a_4+2 \\ =3\times 134+2 \\ =402+2 \\ =404 \end{gathered}$$