Sure, let's find the first five terms of the given recursive sequence step-by-step.
The sequence is defined by the recurrence relation:
[tex]\[ a_n = 3a_{n-1} - 6 \][/tex]
with the initial term:
[tex]\[ a_1 = 7 \][/tex]
Now, we'll calculate each term one by one.
Step 1: Calculate [tex]\( a_1 \)[/tex]
[tex]\[ a_1 = 7 \][/tex]
Step 2: Calculate [tex]\( a_2 \)[/tex]
Using the recurrence relation:
[tex]\[ a_2 = 3a_1 - 6 \][/tex]
Substitute [tex]\( a_1 = 7 \)[/tex]:
[tex]\[ a_2 = 3 \cdot 7 - 6 \][/tex]
[tex]\[ a_2 = 21 - 6 \][/tex]
[tex]\[ a_2 = 15 \][/tex]
Step 3: Calculate [tex]\( a_3 \)[/tex]
Using the recurrence relation:
[tex]\[ a_3 = 3a_2 - 6 \][/tex]
Substitute [tex]\( a_2 = 15 \)[/tex]:
[tex]\[ a_3 = 3 \cdot 15 - 6 \][/tex]
[tex]\[ a_3 = 45 - 6 \][/tex]
[tex]\[ a_3 = 39 \][/tex]
Step 4: Calculate [tex]\( a_4 \)[/tex]
Using the recurrence relation:
[tex]\[ a_4 = 3a_3 - 6 \][/tex]
Substitute [tex]\( a_3 = 39 \)[/tex]:
[tex]\[ a_4 = 3 \cdot 39 - 6 \][/tex]
[tex]\[ a_4 = 117 - 6 \][/tex]
[tex]\[ a_4 = 111 \][/tex]
Step 5: Calculate [tex]\( a_5 \)[/tex]
Using the recurrence relation:
[tex]\[ a_5 = 3a_4 - 6 \][/tex]
Substitute [tex]\( a_4 = 111 \)[/tex]:
[tex]\[ a_5 = 3 \cdot 111 - 6 \][/tex]
[tex]\[ a_5 = 333 - 6 \][/tex]
[tex]\[ a_5 = 327 \][/tex]
Therefore, the first five terms of the sequence are:
[tex]\[ 7, 15, 39, 111, 327 \][/tex]
