Given:
The recursive formulae are:
(a) [tex]f(n)=f(n-1)-2; f(1)=8[/tex]
(b) [tex]f(n)=5+f(n-1); f(1)=0[/tex]
To find:
The explicit equation.
Solution:
A recursive formula of an arithmetic sequence is
[tex]f(n)=f(n-1)+d[/tex] and f(1) is the first term.
Where, d is the common difference.
The explicit formula is
[tex]f(n)=a+(n-1)d[/tex]
where, a is first term and d is common difference.
(a)
We have,
[tex]f(n)=f(n-1)-2; f(1)=8[/tex]
Here, first term is 8 and common difference is -2. So, the explicit formula is
[tex]f(n)=8+(n-1)(-2)[/tex]
[tex]f(n)=8-2n+2[/tex]
[tex]f(n)=10-2n[/tex]
Therefore, the explicit formula is [tex]f(n)=10-2n[/tex].
(b)
We have,
[tex]f(n)=5+f(n-1); f(1)=0[/tex]
Here, first term is 0 and common difference is 5. So, the explicit formula is
[tex]f(n)=0+(n-1)(5)[/tex]
[tex]f(n)=5n-5[/tex]
Therefore, the explicit formula is [tex]f(n)=5n-5[/tex].
