for each recursively defined sequence below write the first few terms. then use the terms to write an explicit equation.

a1 = 17

an+1 = an – 3

a1 = 20

an+1 = 1/2 · an

Explicit formula for

1)

2)

Step-by-step explanation:

W have to find the first few terms of the given sequence and then find an explicit equation

Explicit formula = f(n) + d(n-1) , where,

f(n) is first term,

d is common difference,

n-1 is one term less than the term number.

1)

Given :

Thus, we obtained an explicit formula,

2)

Given :

Thus, we obtained an explicit formula,

Thus, we obtained an explicit formula,

1. Terms are 17, 14, 11, 8, 5,...... and explicit equation is .

2. Terms are 20, 10, 5, 2.5, 1.25,...... and explicit equation is .

Step-by-step explanation:

Ques 1: We are given the recursive formula for the sequence as,

, where .

So, substituting the values of 'n' from {1,2,3,.....}, we get,

Thus, the sequence is given by 17, 14, 11, 8, 5,......

As, the explicit equation of an arithmetic sequence is of the form, , where is the first term and 'd' is the common difference.

As, the common difference, d = 14 - 17 = -3

Thus, we get,

The given sequence has the explicit equation, .

Ques 2: We are given the recursive formula for the sequence as,

, where .

So, substituting the values of 'n' from {1,2,3,.....}, we get,

Thus, the sequence is given by 20, 10, 5, 2.5, 1.25,......

As, the explicit equation of a geometric sequence is of the form, , where is the first term and 'r' is the common ratio.

As, the common ratio,

Thus, we get,

The given sequence has the explicit equation, .