Answered

What are the first five terms of the recursive sequence generated by

[tex]\[
\begin{array}{l}
a_1 = 2 \\
a_n = 3a_{n-1} - 2
\end{array}
\][/tex]

A. [tex]\(2, 4, 10, 28, 82\)[/tex]
B. [tex]\(-2, -8, -26, -80, -242\)[/tex]
C. [tex]\(2, 4, 9, 25, 73\)[/tex]
D. [tex]\(2, 5, 13, 37, 109\)[/tex]



Answer :

GinnyAnswer
To find the first five terms of the recursive sequence defined by:

[tex]\[ \begin{array}{l} a_1=2 \\ a_n=3a_{n-1}-2 \end{array} \][/tex]

we need to calculate each term step-by-step using the initial term and the recurrence relation. Here's the detailed process:

1. First Term:
[tex]\[ a_1 = 2 \][/tex]

2. Second Term:
[tex]\[ a_2 = 3a_1 - 2 \][/tex]
Plugging in [tex]\( a_1 = 2 \)[/tex]:
[tex]\[ a_2 = 3 \cdot 2 - 2 = 6 - 2 = 4 \][/tex]

3. Third Term:
[tex]\[ a_3 = 3a_2 - 2 \][/tex]
Plugging in [tex]\( a_2 = 4 \)[/tex]:
[tex]\[ a_3 = 3 \cdot 4 - 2 = 12 - 2 = 10 \][/tex]

4. Fourth Term:
[tex]\[ a_4 = 3a_3 - 2 \][/tex]
Plugging in [tex]\( a_3 = 10 \)[/tex]:
[tex]\[ a_4 = 3 \cdot 10 - 2 = 30 - 2 = 28 \][/tex]

5. Fifth Term:
[tex]\[ a_5 = 3a_4 - 2 \][/tex]
Plugging in [tex]\( a_4 = 28 \)[/tex]:
[tex]\[ a_5 = 3 \cdot 28 - 2 = 84 - 2 = 82 \][/tex]

Therefore, the first five terms of the sequence are:
[tex]\[ 2, 4, 10, 28, 82 \][/tex]

So the correct answer is:

A. [tex]\(2, 4, 10, 28, 82\)[/tex]