Question

5. For each sequence, write either an explicit or a recursive formula.

a. 1, −1, 1, −1, 1, −1, …

Answer 1

Explicit form is f(n) = (-1)ⁿ⁻¹

Recursive form of sequence is given by f(n) = -f(n-1)

Explanation:

We can see that 1, −1, 1, −1, 1, −1, … is a geometric sequence with common ration -1 and first term 1.

Explicit form is given by

f(n) = arⁿ⁻¹

f(n) = 1 x (-1)ⁿ⁻¹ = (-1)ⁿ⁻¹

Explicit form is f(n) = (-1)ⁿ⁻¹

The explicit form of sequence is given as

f(n) = (-1)ⁿ⁻¹

So nth term is given by

f(n) = (-1)ⁿ⁻¹

(n-1)th term is given by

f(n-1) = (-1)ⁿ⁻²

We have

`(f(n))/(f(n-1))=((-1)^(n-1))/((-1)^(n-2))=(-1)^(n-1-n+2)=-1\\\\f(n)=-f(n-1)`

Recursive form of sequence is given by f(n) = -f(n-1)