The answer is c. 36======analyzing the for-loop, we see that the control variable, k, starts at 0 and increments by after every loop iteration. the condition
![\texttt{k\ \textless \ =7}](/tex.php?f=\texttt{k\ \textless \ =7})
tells us that the loop will stop when k is above 7; in this case, when k is 8.0 1 2 3 4 5 6 7 are the values for k that will be used. each number is for each iteration; therefore, the for-loop will have 8 iterations in total.the code inside the loop itself goes like this: 1st iteration (k=0)starting with sum=0, val=1
![\texttt{sum += val; }](/tex.php?f=\texttt{sum += val; })
means that add
![\texttt{val}](/tex.php?f=\texttt{val})
to
![\texttt{sum}](/tex.php?f=\texttt{sum})
.since sum is 0 and val is 1, 0 + 1 = 1. the new value of sum is 1.
![\texttt{val++; }](/tex.php?f=\texttt{val++; })
means to increment
![\texttt{val}](/tex.php?f=\texttt{val})
by 1.since val is 1, the new value of val of 2.end of iteration with k=0; k is increased by one to become k=12nd iteration (k=1)starting with sum=1, val=2
![\texttt{sum += val; } \iff \texttt{sum = 1 + 2}](/tex.php?f=\texttt{sum += val; } \iff \texttt{sum = 1 + 2})
======let us establish a pattern. this is moreso logic than programming: we can rewrite
![\texttt{sum = 1 + 2}](/tex.php?f=\texttt{sum = 1 + 2})
so that we have the elements from the previous iteration. the 1 came from 0 + 1 before. so we can write it as
![\texttt{sum = 0 + 1 + 2}](/tex.php?f=\texttt{sum = 0 + 1 + 2})
.and if we did the third iteration, by the end, sum would look like
![\texttt{sum = 0 + 1 + 2 + 3}](/tex.php?f=\texttt{sum = 0 + 1 + 2 + 3})
notice how the number of plus signs is equal to the iteration you are on (which is just k+1), and what is being added is increasing by one each time. for example, the third iteration (k=2) ends up with
![\texttt{sum = 0 + 1 + 2 + 3}](/tex.php?f=\texttt{sum = 0 + 1 + 2 + 3})
with 3 plus signs.the second iteration (k=1) ends up with
![\texttt{sum = 0 + 1 + 2}](/tex.php?f=\texttt{sum = 0 + 1 + 2})
with two plus signs.the first iteration (k=0) ends up with
![\texttt{sum = 0 + 1}](/tex.php?f=\texttt{sum = 0 + 1})
with one plus sign.this is a pattern. the nth iteration will have n plus signs.since we established before that there are 8 iterations of this loop in total, the value of sum at k=7 (the 8th and last iteration) will have sum with 8 plus signs.
![\begin{aligned} \texttt{sum} & = \texttt{0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 } \\ & = \texttt{3 + 3 + 4 + 5 + 6 + 7 + 8 } \\ & = \texttt{6 + 4 + 5 + 6 + 7 + 8 } \\ & = \texttt{10 + 5 + 6 + 7 + 8 } \\ & = \texttt{15 + 6 + 7 + 8 } \\ & = \texttt{21 + 7 + 8 } \\ & = \texttt{28 + 8 } \\ & = \texttt{36} \end{aligned}](/tex.php?f=\begin{aligned} \texttt{sum} & = \texttt{0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 } \\ & = \texttt{3 + 3 + 4 + 5 + 6 + 7 + 8 } \\ & = \texttt{6 + 4 + 5 + 6 + 7 + 8 } \\ & = \texttt{10 + 5 + 6 + 7 + 8 } \\ & = \texttt{15 + 6 + 7 + 8 } \\ & = \texttt{21 + 7 + 8 } \\ & = \texttt{28 + 8 } \\ & = \texttt{36} \end{aligned})
so the value of sum at the end of the loop is 36.