Question

3. Given the recursive equation for each arithmetic sequence, write the explicit equation.
f(n)=f(n-1)-2; f(1)=8
f(n)=5+f(n-1);f(1)=0

Answer 1

Given:

The recursive formulae are:

(a) $f(n)=f(n-1)-2; f(1)=8$

(b) $f(n)=5+f(n-1); f(1)=0$

To find:

The explicit equation.

Solution:

A recursive formula of an arithmetic sequence is

$f(n)=f(n-1)+d$ and f(1) is the first term.

Where, d is the common difference.

The explicit formula is

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

where, a is first term and d is common difference.

(a)

We have,

$f(n)=f(n-1)-2; f(1)=8$

Here, first term is 8 and common difference is -2. So, the explicit formula is

$f(n)=8+(n-1)(-2)$

$f(n)=8-2n+2$

$f(n)=10-2n$

Therefore, the explicit formula is $f(n)=10-2n$.

(b)

We have,

$f(n)=5+f(n-1); f(1)=0$

Here, first term is 0 and common difference is 5. So, the explicit formula is

$f(n)=0+(n-1)(5)$

$f(n)=5n-5$

Therefore, the explicit formula is $f(n)=5n-5$.