The document covers sequences, specifically arithmetic and geometric progressions, including definitions, explicit formulas, and examples. It explains summation notation and emphasizes the significance of discrete structures in computer programming, such as arranging applications and data. Several exercises and examples provide practical applications of sequences and their properties.

2. PREPARED BY:
MUHAMMAD ZARYAB

3. SEQUENCE
 Arithmetic sequence / Arithmetic progression
 Geometric sequence / Geometric progression
SUMMATION
Sigma notation

8. SEQUENCE:
A sequence is just a list of elements usually written in a row.
EXAMPLES:
1. 1, 2, 3, 4, 5, …
2. 4, 8, 12, 16, 20,…
3. 2, 4, 8, 16, 32, …
4. 1, 1/2, 1/3, 1/4, 1/5, …
5. 1, -1, 1, -1, 1, -1, …
Find the first 4 terms
C n = 1+ (-1) ^n for all integers n ≥ 0
a n = (-1) n for all integers n ≥ 0
bj = 1 + 2j, for all integers j ≥ 0
An=n+1/2n+5 for all integers n ≥ 0

9. Find the explicit formula
1. 1-1/2 , 1/2-1/3 , 1/3-1/4
2. 2, 6, 12, 20, 30, 42, 56, …
3. 1/4, 2/9, 3/16, 4/25, 5/36, 6/49, …
Find the specified term of A.P
1)3,7,11......61 term
2)6,4,2....45 term
3)-4,-7,-10,.....19 term

10. Find the term indicated in following sequence
1) 2,6,11,17....a8
2) 2)1,1/3,1/9,1/27...a5
Find the missing element of A.P
a=2,an=402 ,n=26
A=16, an=0 ,d=-1/4
Find the 15th term of A.P where 3rd term is 8 and
common difference is 1/8;
Which term of A.P is 6,2,-2,..........is -146
Find the nth term of A.P where An-5=3n+9
Find the nth term of A.P (3/4)2 , (3/7)2 , (3/10)2

11. Find A.M between X2 +X+1 , X2 -X+1 ;
If 3 and 6 are two A.Ms between a and b ,find A and B;
Find 6 A.Ms between 7 and 13;
If the A.M 9 and B is 27 ,find the valve of B;

12. GEOMETRIC SEQUENCE:
A sequence in which every term after the first is
obtained from the preceding term by multiplying it
with a constant number is called a geometric sequence
or geometric progression (G.P.)
the common ratio of the G.P. commonly denoted by “r”.
EXAMPLE:
1. 1, 2, 4, 8, 16, … (common ratio = 2)
2. 3, - 3/2, 3/4, - 3/8, … (common ratio = - 1/2)
3. 0.1, 0.01, 0.001, 0.0001, … (common ratio = 0.1 = 1/10)

15. Find the 8th term of the following geometric sequence 4, 12, 36, 108, …
Which term of the geometric sequence is 1/8 if the first term is 4 and
common ratio ½
Write the geometric sequence with positive terms whose second term is 9
and fourth term is 1
Write the geometric sequence with positive terms whose second term is 9
and fourth term is 1.
Find the missing term
R=10,an=100,a=1;
A=128 ,r=1/2 ,an=1/4
If the three A.Ms between A and B are 5,9,13 Find A and
B.

18. SUMMATION NOTATION:
The capital Greek letter sigma ∑ is used to write a sum in
a short hand notation. where k varies from 1 to n
represents the sum given in expanded form by = a1 +
a2 + a3 + … + an

19. How important the subject
discrete structure in computer
programming ?
 Used to arrange Apps(Application) in any sequence
 Used to arrange your Phone contacts in any sequence
 Discrete Structure gives you complete structure about
computer programming
 Without discrete structure you don’t gave any sequence
in your programming.
 Ask examples , like arranging array, char array

20. THANK YOU