State whether the sequence is arithmetic or geometric, write a recursive and explicit formula and find the 10th term.

81, -27, 9, -3, 1...
Sequence Type: geometric
Recursive: _______
Explicit: ______
10th Term: -1/243

recursive: it just describes the pattern

"to get a new term, multiply the previous by -1/3

so term(n) = (-1/3)term(n-1)

explicit:
t(n) = a r^(n-1) <---- which is your formula for the general term
We have a = 81 and r = -1/3

t(n) = 81(-1/3)^(n-1) , but 81 = 3^4 or (-3)^4 AND (-1/3)^(n-1) = (-3)^(1-n)

t(n) = (-3)^4 * (-3)^(1-n)
= (-3)^(5-n)


check one of them:
t(3) = (-3)^2 = 9
Check some others if you wish

eg. term(10) = (-3)^-5
= 1/(-3)^5 = -1/243

To determine whether a sequence is arithmetic or geometric, we need to check whether there is a common difference (arithmetic) or a common ratio (geometric) between consecutive terms.

In the given sequence 81, -27, 9, -3, 1..., we can observe that each term is obtained by multiplying the previous term by -1/3. Therefore, the sequence is geometric.

The recursive formula for a geometric sequence is derived by expressing each term in terms of the previous term. Let's denote the nth term as a(n). In this case, we can derive the recursive formula as follows:

a(1) = 81 (the first term)
a(n) = a(n-1) * (-1/3) (for n > 1)

Now let's derive the explicit formula for the given geometric sequence. The explicit formula allows us to directly find the nth term of the sequence without calculating the previous terms recursively.

In a geometric sequence, the nth term (a(n)) can be found using the formula:

a(n) = a(1) * r^(n-1)

where a(1) is the first term and r is the common ratio.

In this case, a(1) = 81 and r = -1/3. Hence, the explicit formula for the given sequence is:

a(n) = 81 * (-1/3)^(n-1)

Now, let's find the 10th term of the sequence using the explicit formula:

a(10) = 81 * (-1/3)^(10-1)
= 81 * (-1/3)^9
= -1/243

Therefore, the 10th term of the given geometric sequence is -1/243.