Answer: Q(n) = Q(n - 1) + 2.5
Explanation:
We have 3 values of the sequence Q(n)
These values are:
Q(1) = 3
Q(3) = 8
Q(7) = 18
I would think that this is a geometric sequence.
Remember that the equation for the n-th term of a geometric sequence is:
A(n) = A(1)*r^(n-1)
where r is a constant, and A(1) is the first term of the sequence.
If we rewrite the terms that we know of Q(n) in this way we get:
Q(3) = Q(1)*r^(3 - 1) = 3*r^2 = 8
Q(7) = Q(1)*r^(7 - 1) = 3*r^6 = 18
Then we have two equations:
3*r^2 = 8
3*r^6 = 18
We should see if r is the same for both equations:
in the first one we get:
r^2 = 8/3
r = (8/3)^(1/2) = 1.63
and in the other equation we get:
r^6 = 18/3
r = (18/3)^(1/6) = 1.34
Then this is not a geometric sequence.
Now let's see if this is an arithmetic sequence.
The n-th term of an arithmetic sequence is written as:
A(n) = A(1) + (n - 1)*d
where d is a constant.
If we write the terms of Q(n) that we know in this way we get:
Q(3) = Q(1) + (3 - 1)*d = 3 + 2*d = 8
Q(7) = Q(1) + (7 - 1)*d = 3 + 6*d = 18
We need to see if d is the same value for both equations.
in the first one we get:
3 + 2*d = 8
2*d = 8 - 3 = 5
d = 5/2 = 2.5
In the second equation we get:
3 + 6*d = 18
6*d = 18 - 3 = 15
d = 15/6 = 2.5
d is the same for both terms, then this is an arithmetic sequence.
An arithmetic sequence is a sequence where the difference between any two consecutive terms is always the same value (d)
Then the recursive relation is written as:
A(n) = A(n - 1) + d
Then the recursive relation for Q is:
Q(n) = Q(n - 1) + 2.5