Question

Create an explicit equation for each recursively-defined sequence below.

a. a1=17, an+1 = an-7
b. t(1)=3, t(n+1)=t(n)+6

Answer 1

The explicit equations are a(n) = 24 - 7n and t(n) = -3 + 6n

Creating an explicit equation for the recursively-defined sequence

From the question, we have the following parameters that can be used in our computation:

a1 = 17,

a(n + 1) = an - 7

Here, we have the first term to be

a = 17

The common difference is the term added or subtracted to a(n)

So, we have

d = -7

An arithmetic sequence is represented as

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

So, we have

a(n) = 17 + (n - 1) * -7

a(n) = 17 - 7n + 7

a(n) = 24 - 7n

Next, e have

t(1) = 3,

t(n+1)=t(n)+6

Using the same step as (a), we have

t(n) = 3 + (n - 1) * 6

t(n) = 3 + 6n - 6

t(n) = -3 + 6n

Hence, the explicit equations are a(n) = 24 - 7n and t(n) = -3 + 6n