find the first five terms of {2n/3 n+1} a1=-2 an+3an-1?

To find the first five terms of the sequence {2n/3 n+1} with the given recurrence relation a1 = -2 and an+3 = an - 1, we can use the formula for the nth term of the sequence.

Let's calculate the first five terms step by step:

Term 1 (a1):
Given that a1 = -2, the first term of the sequence is -2.

Term 2 (a2):
To find the second term, substitute n = 2 into the general formula.
a2 = (2(2))/((3(2)) + 1) = 4/7

Term 3 (a3):
To find the third term, substitute n = 3 into the general formula.
a3 = (2(3))/((3(3)) + 1) = 6/10 = 3/5

Term 4 (a4):
To find the fourth term, we use the recurrence relation.
a4 = a1 + 3a3
Substituting the previously calculated terms:
a4 = -2 + 3 * (3/5) = -2 + 9/5 = 1/5

Term 5 (a5):
To find the fifth term, we use the recurrence relation again.
a5 = a2 + 3a4
Substituting the previously calculated terms:
a5 = 4/7 + 3 * (1/5) = 4/7 + 3/5 = 20/35 + 21/35 = 41/35

Therefore, the first five terms of the sequence {2n/3 n+1} with the given recurrence relation a1 = -2 and an+3 = an - 1 are:
-2, 4/7, 3/5, 1/5, 41/35