Question

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=

Answer 1

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