Theory in practical computing scenarios PDF

LO1 Use applied number theory in practical computing scenarios
Part 1 Number theory
Part 2 sequences and Series
2.1
¿findthe∑of allnaturalnumbersbetween1∧1000whicharedivisibleby7,wefirstdeterminethe
largestmultipleof 7lessthan1000.Letthenumberbek.
k=7∟ 1000
7⅃=994
k=994
Therefore,∑of allnaturalnumbersbetween1∧1000willbegivenby:
7(1+2+...+994)=7+14+21+...+994
¿7∗142∗143
2
¿71071
Hence the sum of all natural numbers between 1 and 10000 which are divisible by 7 is 71071.
2.2
If a ball is dropped vertically from a tower 3.6 meters high and the height of its rebound is
recorded for 4 successive bounces as shown in in the table below, the heights of the bouncing ball
shown in the table form a geometric sequence. A geometric sequence is a sequence of numbers in
which the next is found by multiplying the preceding term by a common term[ CITATION Sea12 \l 1033 ].
Generally,
lettingthenthterm¿bean=a1rnCITATION Kei11¿1033(Hirst,2011);
wherea1isthe firstterm,risthecommonterm(samevalue)
¿thetablebelow,thefollowingcomputationscanbecarried.
270
360=3
4
202.5
270=3
4

151.875
202.5=3
4
Consequently,r=3
4∧a1=360
Usingthegeneralformulaeabove,270=360∗(3
4)2−1
=270
Therefore, the results in the table above demonstrate a geometric sequence.
TheHeightof the4th Bounce=360(3
4)4
=360∗(3
4)4
=113.90625cm
The expression for the height of the nth bounce will be given by
an=a1rn
Theheightof the15thbounce=360(3
4)15
=4.810845964cm
The∑of allheightsdroppedbytheballtillitcomes¿astandstillis givenby,
∑i=0
n
ai=a0(1−rn
1−r )=360
(1−(3
4)∞
1−3
4)=360
(1
1
4)=360∗4
¿1440cm
Prime numbers has a number of applications in the field of computing. Prime can be applied
public key cryptography that is used to encrypt (protect) electronic data[ CITATION Ash17 \l 1033 ]. For
example, suppose we want a function that takes 2 numbers h and k and gives a new number in a way
that h and k can be recovered from output of the function. All we need to do is multiply the hth prime
number and kth prime number. The product is the third number we are looking for. The advantage of
prime numbers employed in data encryption is uniqueness[ CITATION Car12 \l 1033 ]. Though at
advancement level, banking systems use the same model to protect their data. Besides, in public key
cryptography, prime numbers are used to communicate. For example, security officers use encrypted
information to prevent the enemy accessing the messages being shared[ CITATION Gra15 \l 1033 ].