Given:
The different recursive formulae.
To find:
The explicit formulae for the given recursive formulae.
Solution:
The recursive formula of an arithmetic sequence is
and the explicit formula is
, where a is the first term and d is the common difference.
The recursive formula of a geometric sequence is
and the explicit formula is
, where a is the first term and r is the common ratio.
The first recursive formula is:
![f(1)=5](https://tex.z-dn.net/?f=f%281%29%3D5)
for
.
It is the recursive formula of an arithmetic sequence with first term 5 and common difference 5. So, the explicit formula for this recursive formula is:
![f(n)=5+(n-1)(5)](https://tex.z-dn.net/?f=f%28n%29%3D5%2B%28n-1%29%285%29)
![f(n)=5+5(n-1)](https://tex.z-dn.net/?f=f%28n%29%3D5%2B5%28n-1%29)
Therefore, the correct option is A, i.e.,
.
The second recursive formula is:
![f(1)=5](https://tex.z-dn.net/?f=f%281%29%3D5)
for
.
It is the recursive formula of a geometric sequence with first term 5 and common ratio 3. So, the explicit formula for this recursive formula is:
![f(n)=5(3)^{n-1}](https://tex.z-dn.net/?f=f%28n%29%3D5%283%29%5E%7Bn-1%7D)
Therefore, the correct option is F, i.e.,
.
The third recursive formula is:
![f(1)=5](https://tex.z-dn.net/?f=f%281%29%3D5)
for
.
It is the recursive formula of an arithmetic sequence with first term 5 and common difference 3. So, the explicit formula for this recursive formula is:
![f(n)=5+(n-1)(3)](https://tex.z-dn.net/?f=f%28n%29%3D5%2B%28n-1%29%283%29)
![f(n)=5+3(n-1)](https://tex.z-dn.net/?f=f%28n%29%3D5%2B3%28n-1%29)
Therefore, the correct option is D, i.e.,
.