The 5th term in a sequence is 27, and each term is 3 less than the previous term. Enter an explicit rule and a recursive rule to complete the description of the sequence.

Recursive Rule = > a (n+1) = a (n) - 3

Explicit Rule = > a (n) = 15 - (n-1) * 3

Sequence = 39, 36, 33, 30, 27

Explanation:

Recursive Rule:

Let a (n) denotes nth term and a (n-1) denotes the previous term

We are given the each successive term is 3 less than the previous term so we can write

a (n) = a (n-1) - 3

we can also write

Recursive Rule = > a (n+1) = a (n) - 3

It still means the same. Each new term is 3 less than the previous term.

Explicit Rule:

We are also given that the 5th term a (5) is equal to 27 and we already know the difference between successive terms that is - 3

a (n) = a (1) + (n-1) * d

We need the first term to specify the explicit rule

For the 5th term we can write

a (5) = a (1) + (5-1) * -3

27 = a (1) - (4) * 3

27 = a (1) - 12

a (1) = 27 + 12 = 39

Explicit Rule = > a (n) = 15 - (n-1) * 3

Verification:

a (1) = 39

a (2) = a (1) - (2-1) * 3 = 39 - (1) * 3 = 39 - 3 = 36

a (3) = a (1) - (3-1) * 3 = 39 - (2) * 3 = 39 - 6 = 33

a (4) = a (1) - (4-1) * 3 = 39 - (3) * 3 = 15 + 9 = 30

a (5) = a (1) - (5-1) * 3 = 39 - (4) * 3 = 15 + 12 = 27

sequence = 39, 36, 33, 30, 27

a (n) = a (n-1) - 3

a (5) = a (5-1) - 3 = a (4) - 3 = 30 - 3 = 27

a (4) = a (4-1) - 3 = a (3) - 3 = 33 - 3 = 30

a (3) = a (3-1) - 3 = a (2) - 3 = 36 - 3 = 33

a (2) = a (2-1) - 3 = a (1) - 3 = 39 - 3 = 36

a (1) = 39

sequence = 39, 36, 33, 30, 27

Conclusion:

Hence as you can see both of the rules Recursive and Explicit have being tested and verified. The obtained results are correct.