Use the initial term and the recursive formula to find an explicit formula for the sequence an. Write your answer in simplest form. a1 = 28 an = an–1–5 an=
To find the explicit formula for the sequence, we can start by listing out the first few terms of the sequence:
a1 = 28
a2 = a1 - 5 = 28 - 5 = 23
a3 = a2 - 5 = 23 - 5 = 18
a4 = a3 - 5 = 18 - 5 = 13
a5 = a4 - 5 = 13 - 5 = 8
From observing the terms, we can notice that each term is 5 less than the previous term. We can express this observation using the recursive formula:
an = an-1 - 5
To find the explicit formula, we start by determining a general expression for an in terms of n. Let's assume the explicit formula is of the form:
an = Cn + D
where C and D are constants to be determined.
From the recursive formula, we know that:
a2 = a1 - 5 = C(2) + D = 23
a1 = 28. Therefore:
C(1) + D = 28
So, D = 28 - C
Substituting D in terms of C in our equation for a2:
C(2) + 28 - C = 23
2C + 28 - C = 23
C + 28 = 23
C = -5
Substituting C = -5 in the equation D = 28 - C:
D = 28 - (-5)
D = 28 + 5
D = 33
So, the explicit formula for the sequence an is:
an = -5n + 33