Problem $3$: Below are two functions that calculates the sum of the first $n$

integers using python. In other words, ${\displaystyle \underset{?}{\overset{?}{}}}(n)=1+2+3+$dots$+n.$

def $\}\backslash$operatorname$\{$sum$\}1($n$)$

if $(\}$n$==1)\backslash$mathrm$\{$ :

return $1$

else:

return n$+$ sum$1($n$-1)$

def sum$2($n$)$:

return n$*($n$+1)/2$

Answer the questions below:

$($a$)$ Choose $5$ different numbers for $n.$ Verify that sum $1(n)$ equals sum $2(n)$ for

these $5$ numbers.

$($b$)$ Which of these two functions do you think is generally more efficient for

$n>100.$ Why do you think this is so$?$

$($c$)$ Do you think sum$1(1)$ or sum $2(1)$ is more efficient? Why?

$($A$)$ What happens if you call sum$1(0)?$

$($e$)$ Let's define the open sentance:

$\{\begin{array}{ccc}\text{:}P(n)& \text{:}\text{s}\text{u}\text{m}\text{l}1(n)is\text{a}\text{v}\text{a}\text{l}\text{i}\text{d}\text{c}\text{a}\text{l}\text{c}\text{u}\text{l}\text{a}\text{t}\text{i}\text{o}\text{n}\text{.}& \end{array}$

i$.$ What is the smallest ninN such that $P(n)$ is true? Let's label this

case $P(b),$ where $b$ is this smallest such number. In other words,

what is $b?$

true statements?

iii. Is the following statement true or false: $P(b)=>P(b+1)?$

iv$.$ Assume you could use $\frac{{\displaystyle \underset{?}{\overset{?}{}}}2(n)}{?}$ to validate whether sum$1($n$)$ is correct

for large number of $n.$ How would you rewrite the open statement

$P(n)?$

v$.$ Can you rewrite sum$1($n$)$ so that it calls sum $2(n)?$