Let's start by analyzing the given sequence defined by the recurrence relation [tex]\( a_{n+1} = 2a_n \)[/tex] with the initial term [tex]\( a_1 = 3 \)[/tex].
1. Finding [tex]\( a_2 \)[/tex]:
We know that [tex]\( a_2 \)[/tex] is the next term after [tex]\( a_1 \)[/tex]. According to the recurrence relation, [tex]\( a_{n+1} = 2a_n \)[/tex]. For [tex]\( n = 1 \)[/tex],
[tex]\[
a_2 = 2a_1
\][/tex]
Substituting [tex]\( a_1 \)[/tex] with 3,
[tex]\[
a_2 = 2 \cdot 3 = 6
\][/tex]
2. Finding [tex]\( a_3 \)[/tex]:
Now, we find [tex]\( a_3 \)[/tex], which is the term following [tex]\( a_2 \)[/tex]. Again, using the recurrence relation [tex]\( a_{n+1} = 2a_n \)[/tex]. For [tex]\( n = 2 \)[/tex],
[tex]\[
a_3 = 2a_2
\][/tex]
Substituting [tex]\( a_2 \)[/tex] with the value we just found (6),
[tex]\[
a_3 = 2 \cdot 6 = 12
\][/tex]
Therefore, the values of [tex]\( a_2 \)[/tex] and [tex]\( a_3 \)[/tex] are [tex]\( 6 \)[/tex] and [tex]\( 12 \)[/tex] respectively.
