HiLCoE
School of Computer Science and Technology
CS221: Computer Programming I (Using C++)
Autumn 2015
Worksheet I
1. Don’t you love triangles?
A number N is called “triangular” if you can take N little
asterisks and make an equilateral triangle from them. For
example, here are the first few triangular numbers:
*
* *
3
*
* *
* * *
6
*
* *
* * *
* * * *
10
*
* *
* * *
* * * *
* * * * *
15
*
* *
* * *
* * * *
* * * * *
* * * * * *
21
Here
is
an
algorithm
that
is
attempting
to
detect
triangularity in the input. But it has a bug. Fix the
algorithm and draw the flowchart for the algorithm you fixed.
START
OUTPUT “Enter a positive integer”
INPUT N
i = 1
S = 0
WHILE (S < N)
i = i + 1
S = S + i
END WHILE
IF (S = N)
OUTPUT “Triangular”
ELSE
OUTPUT “Not triangular”
END IF
STOP
HiLCoE
School of Computer Science and Technology
CS221: Computer Programming I (Using C++)
Autumn 2015
2. The oscillating sign
Write an algorithm in pseudo-code that accepts a positive
integer N and outputs a 1(one) if N is even and -1 (minus one)
if N is odd. Hint: -1 multiplied by itself an even number of
times is +1, while -1 multiplied by itself an odd number of
times is -1.
3. De-Gauss this!
Gauss’s formula for the computation of the Nth triangular
number is given as:
1 + 2 + 3 + ... + N
=
N(N + 1)
2
Here is another algorithm attempting to detect triangularity
of the input. Do you think this one is OK? If not, fix it.
START
OUTPUT “Enter a positive integer”
INPUT N
i = 1
WHILE [(i × (i + 1))/2 < N]
i = i + 1
END WHILE
IF [(i × (i + 1))/2 = N]
OUTPUT “Not triangular”
ELSE
OUTPUT “Triangular”
END IF
STOP
HiLCoE
School of Computer Science and Technology
CS221: Computer Programming I (Using C++)
Autumn 2015
4. Here is an algorithm in pseudo-code:
START
OUTPUT “Enter a positive integer”
INPUT N
i = 1
WHILE (i ×i < N)
i = i + 1
END WHILE
IF (i × i = N)
OUTPUT “YES”
ELSE
OUTPUT “NO”
END IF
STOP
For which of the following inputs will the algorithm say
“YES”?
(a)
(b)
(c)
(d)
28
36
54
81
What property of the input is the algorithm determining?
5. Summing the squares
Describe an algorithm in either pseudo-code or flowchart form,
to sum the squares o f the integers up to and including some
N, where N is an input to the algorithm. That is,
S = 12 + 22 + 32 + · · · + N2
Where S is the sum. You must express your algorithm in terms
of a loop that is traversed N times. (There is a closed
mathematical expression S = N (N + 1) (2N + 1) / 6 that you
must not exploit in your algorithm. Pretend the closed
mathematical expression does not exist.) Make sure to declare
and initialize all the variables you use.
HiLCoE
School of Computer Science and Technology
CS221: Computer Programming I (Using C++)
Autumn 2015
6. Summing an oscillating series
Describe an algorithm in both pseudo-code and flowchart form,
to perform the following sum
S = 1− x + x2 − x3 + x4 · · · +(-x)N
Where x and N are inputs and S is the sum, an output. You must
express your algorithm in terms of a loop that is traversed N
times. Express your algorithm in terms of pseudo-code and a
flowchart.
7. Solve for x:
ax2 + bx + c = 0
where a, b, c are arbitrary constants
The quadratic equation has real solutions:
x = −b ± √b2 – 4ac
if a = 0, and b2 – 4ac ≥ 0
2a
Under some circumstances, (a = 0, b2 – 4ac > 0), we have two
solutions:
x = −b ± √b2 – 4ac
2a
2
What if a = 0, b – 4ac = 0? We have one solution:
x = −b
2a
What if b2 – 4ac < 0? No real solution.
What if a = 0? Re-analyze!
bx + c = 0 (y = bx + c is an equation for a straight line!)
x = −c
N.B. b = 0
b
This is the place where a straight line crosses the x-axis.
Give the solution to the quadratic equation both in pseudocode and flow chart.