Find the first five terms of the given recursively defined sequence.
an = 3an − 1 + 9 and a1 = 2
If yourexpression mean :
an = 3 a ( n − 1 ) + 9
a1 = 2
then
a2 = 3 a1 + 9 = 3 * 2 + 9 = 6 + 9 = 15
a3 = 3 a2 + 9 = 3 * 15 + 9 = 45 + 9 = 54
a4 = 3 a3 + 9 = 3 * 54 + 9 = 162 + 9 = 171
a5 = 3 a4 + 9 = 3 * 171 + 9 = 513 + 9 = 522
To find the first five terms of the given recursively defined sequence, we can use the recursive formula provided and iterate through it step by step.
Given: an = 3an − 1 + 9 and a1 = 2
Step 1: Finding the second term (a2):
Using the recursive formula, we substitute n = 2:
a2 = 3a2 − 1 + 9
= 3a1 + 9
= 3(2) + 9
= 6 + 9
= 15
So, the second term (a2) is 15.
Step 2: Finding the third term (a3):
Using the recursive formula, we substitute n = 3:
a3 = 3a3 − 1 + 9
= 3a2 + 9
= 3(15) + 9
= 45 + 9
= 54
So, the third term (a3) is 54.
Step 3: Finding the fourth term (a4):
Using the recursive formula, we substitute n = 4:
a4 = 3a4 − 1 + 9
= 3a3 + 9
= 3(54) + 9
= 162 + 9
= 171
So, the fourth term (a4) is 171.
Step 4: Finding the fifth term (a5):
Using the recursive formula, we substitute n = 5:
a5 = 3a5 − 1 + 9
= 3a4 + 9
= 3(171) + 9
= 513 + 9
= 522
So, the fifth term (a5) is 522.
Therefore, the first five terms of the given recursively defined sequence are:
a1 = 2, a2 = 15, a3 = 54, a4 = 171, a5 = 522.
To find the first five terms of the given recursively defined sequence, we need to apply the recursive formula to each term, starting with the initial term.
Given the recursive formula:
an = 3an − 1 + 9
Starting with the initial term a1 = 2, we can substitute it into the recursive formula to find a2:
a2 = 3a1 - 1 + 9
a2 = 3(2) - 1 + 9
a2 = 6 - 1 + 9
a2 = 14
Now, we can substitute a2 into the recursive formula to find a3:
a3 = 3a2 - 1 + 9
a3 = 3(14) - 1 + 9
a3 = 42 - 1 + 9
a3 = 50
Continuing this process, we can find the next terms:
a4 = 3a3 - 1 + 9
a4 = 3(50) - 1 + 9
a4 = 150 - 1 + 9
a4 = 158
a5 = 3a4 - 1 + 9
a5 = 3(158) - 1 + 9
a5 = 474 - 1 + 9
a5 = 482
Therefore, the first five terms of the given recursively defined sequence are:
a1 = 2, a2 = 14, a3 = 50, a4 = 158, a5 = 482