## Purpose

In this lab you will explore sequences $($arithmetic and geometric$)$ by implementing explicit and recurrence formulas using for loops and recursive functions.

## Code Block

Look over the code below. It is written in C not C$++$ but it should be fairly straightforward to read if you know C$++.$ Answer the questions below and write the code. I will refer to the for statments by the order in which they appear in the code for instance block one refers to the first for loop.

#include

int a$($int a$0,$ int d$,$ int i$)$

$\{$

return d$*$i $+$ a$0$;

$\}$

int recurrencea$($int anminusone, int d$)$

$\{$

return anminusone $+$ d;

$\}$

int fibb$($int n$)$

$\{$

if$($n $==0||$ n $==1)\{$

return $1$;

$\}$

return n$*$fibb$($n$-1)$;

$\}$

int main$()$

$\{$

for$($int i$=0$; i$<=5$; i$++)\{$

printf$("\%$d

$",$ i$)$;

$\}$

printf$("$

$")$;

for$($int i$=0$; i$<=5$; i$++)\{$

printf$("\%$d

$",$a$(17,3,$ i$))$;

$\}$

printf$("$

$")$;

int a $=17$;

printf$("\%$d

$",$ a$)$;

for$($int i$=0$; i$<5$; i$++)\{$

printf$("\%$d

$",$ recurrencea$($a$,3))$;

a $=$ recurrencea$($a$,3)$;

$\}$

printf$("$

$")$;

for$($int i$=0$; i$<6$; i$++)\{$

printf$("\%$d

$",$ fibb$($i$))$;

$\}$

return $0$;

$\}$

$0$

$1$

$2$

$3$

$4$

$5$

$17$

$20$

$23$

$26$

$29$

$32$

$17$

$20$

$23$

$26$

$29$

$32$

$1$

$1$

$2$

$6$

$24$

$120$

## Questions

$1.$ Blocks $2$ and $3$ generate the same output but their $`$for$`$ loops are not the same. How are the loops different and why? Could you rewrite block $3$ so that the bounds of the loop are the same as in block $2?$

$1.$ Implement a $`$for$`$ loop which generates and prints the $16$th through $27$th terms of the geometric sequence that starts at $18\mathrm{.}25$ and has a common ratio of $1\mathrm{.}5.$

$1.$ Implement a function that uses a recurrence relation to achieve the same as above.

$1.$ Write a short program which stores the $3$rd through $99$th terms of the arithmetic sequence which starts at $54321$ and has a common difference of $-33\mathrm{.}2$ in an array. Generate the terms using a function that implements a recurrence relation using the previous element of the array.

$1.$ Write a short program using a recursive function that generates the terms of the sequence a$\_$n $=$ a$\_$n$-1*$ a$\_$n$-2.$ Generate the first twenty terms of the sequence if a$\_1=3$ and a$\_2=0\mathrm{.}5.$ Store the elements in an array. Print the elements in the array.