Answer:
The answer is:
a^n = 2 + 6(n-1)
Explanation:
Find the first few terms of the sequence.
We know that a
0
=2, so the first term of the sequence is 2.
From the given recurrence relation, we can find the next few terms as follows:
a^1 = 6 + a^0 = 6 + 2 = 8
a^2 = 6 + a^1 = 6 + 8 = 14
a^3 = 6 + a^2 = 6 + 14 = 20
Look for a pattern in the sequence.
We can see that the sequence is increasing by 6 each time. This suggests that the sequence is an arithmetic sequence with a common difference of 6.
Find the general formula for the sequence.
The general formula for an arithmetic sequence is a
n
=a
1
+d(n−1), where a
1
is the first term and d is the common difference.
In this case, the first term is 2 and the common difference is 6, so the general formula for the sequence is a
n
=2+6(n−1).
Answer the question.
The question asks us to find the value of a
n
for any given value of n. We can use the general formula to do this.
For example, to find the value of a
4
, we can simply substitute n=4 into the general formula:
a^4 = 2 + 6(4-1) = 2 + 6(3) = 20
Therefore, the value of a
n
for any given value of n is a
n
=2+6(n−1).