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![Geometric Progression
A sequence in which every term is obtained by multiplying or dividing a definite number with the
preceding number is known as a geometric sequence.
𝑟 =
𝑎 𝑛
𝑎 𝑛−1
( 𝑛 > 1), ∀𝑛 ∈ 𝑁, 𝑎 𝑛−1 ≠ 0
Example:
Q: Find 𝑎 𝑛 if 𝑎4 =
8
27
and 𝑎7 −
64
729
of a G.P.
Sol:
Since 𝑎4 =
8
27
𝑎1 𝑟3
=
8
27
----------(1)
Also 𝑎7 = −
64
729
𝑎1 𝑟6
= −
64
729
----------(2)
Dividing eq (1) from (2), we get
𝑎1 𝑟6
𝑎1 𝑟3 =
−
644
729
8
27
𝑟3
= −
64
729
×
27
8
𝑟3
= − (
2
3
)
3
, real root only.
𝑟 = −
2
3
Put 𝑟 = −
2
3
in eq (1)
𝑎1 [−
2
3
]
3
=
8
27
−
8
27
𝑎1 =
8
27
𝑎1 = −1
𝑎 𝑛 = 𝑎1 𝑟 𝑛−1
𝑎 𝑛 = (−1)[−
2
3
]
𝑛−1
𝑎 𝑛 = (−1) 𝑛
[
2
3
]
𝑛−1](https://image.slidesharecdn.com/dsass-200304170157/75/Sequence-and-Series-Word-File-Discrete-Structure-4-2048.jpg)
Uploaded byZain Abid
Sequence and Series Word File || Discrete Structure
The document discusses the concept of sequences and series, highlighting their applications in various fields such as physics and computer science. It explains different types of sequences, including arithmetic, geometric, and harmonic progressions, along with their means and provides examples for clarity. Additionally, it distinguishes between sequences and series, emphasizing the significance of order in sequences and summation in series.
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Sequences and series have applications in various fields. A sequence is an ordered arrangement, essential in programming algorithms.
Topics include Arithmetic Progression (A.P), Arithmetic Mean (A.M). A.P is formed by adding a common difference. Example shows calculation of 10th term.
Geometric Progression (G.P) is where terms are multiplied or divided by a fixed number. Example provides calculations leading to the nth term.
Geometric Mean (G.M) connects two numbers in G.P. Harmonic Progression (H.P) is derived from A.P of reciprocals. Harmonic Mean example is provided.
Sequences represent ordered lists. A structured form aids in understanding and advising across disciplines. Differences between sequences and series are highlighted.
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Similar to Sequence and Series Word File || Discrete Structure
Sequence and Series Word File || Discrete Structure
- 1. Introduction The sequence andseries have important applications in Physics, Geography, Biology, Economics, Psychology, Engineering, Chemistry, Computer science and Finance. Sequence A sequence is an arrangement of any objects or a set of numbers in a particular order followed by some rule. If a1, a2, a3, a4, ……… etc. Denote the terms of a sequence, then 1,2,3,4, … Example: 25,21,17,13,9, …. 𝑎1 = 25 𝑎2 = 25 + (−4) = 21 𝑎3 = 21 + (−4) = 17 𝑎4 = 17 + (−4) = 13 𝑎5 = 13 + (−4) = 9 𝑎6 = 9 + (−4) = 5 𝑎7 = 5 + (−4) = 1 Hence Next two terms are: 5 & 1 Sequence in computer science (programming) In Programming, sequence is the basic algorithm: A set of logical steps carried out in order that commands are executed by a computer, allows us to carry out tasks that have multiple steps. Microprocessor also follow a sequence which is Fetch, Decode, and then Execute.
- 2. Topics: - ArithmeticProgression (A.P) Arithmetic Mean (A.M) Geometric Progression (G.P) Geometric Mean (G.M) Harmonic Progression (H.P) Harmonic Mean (H.M) Arithmetic Progression An Arithmetic Progression is a sequence in which every term is obtained by adding some fix number to the preceding term. The fix number to preceding term. The fix number is the common difference of two consecutive terms denoted by d. Example: Q: Find the 10th term if 𝑎3 = 8 and 𝑎7 = 16 Sol: Since 𝑎3 = 8 For 3rd term: 𝑎1 + 2𝑑 = 8 ----------(1) Also 𝑎7 = 16 𝑎1 + 6𝑑 = 16 ----------(2) Subtract eq (1) from (2), we get 𝑎1+6𝑑 =16 𝑎1+2𝑑=8 4𝑑=8 (Subtract) 𝑑 = 2 Put 𝑑 = 2 in eq (1) 𝑎1 + 2(2) = 8 𝑎1 + 4 = 8 𝑎1 = 8 − 4 𝑎1 = 4
- 3. Now 𝑎10 = 𝑎1+ 9𝑑 𝑎10 = 4 + 9(2) 𝑎10 = 4 + 18 𝑎10 = 22 Arithmetic Mean Let a and b be two numbers, than a number A is set to be A.M between a and b if a, A, b are in A.P Thus A - a = b – A 2A =a +b A = a+b/2 Example: Q: Find A.M between 2𝑥 + 1, 4𝑥 + 3 Sol: 2𝑥 + 1, 4𝑥 + 3 Here 𝑎 = 2𝑥 + 1 , 𝑏 = 4𝑥 + 3 𝐴 = 𝑎+𝑏 2 𝐴 = 2𝑥+1+4𝑥=3 2 𝐴 = 6𝑥+4 2 𝐴 = 2(3𝑥+3) 2 𝐴 = 3𝑥 + 2
- 4. Geometric Progression A sequencein which every term is obtained by multiplying or dividing a definite number with the preceding number is known as a geometric sequence. 𝑟 = 𝑎 𝑛 𝑎 𝑛−1 ( 𝑛 > 1), ∀𝑛 ∈ 𝑁, 𝑎 𝑛−1 ≠ 0 Example: Q: Find 𝑎 𝑛 if 𝑎4 = 8 27 and 𝑎7 − 64 729 of a G.P. Sol: Since 𝑎4 = 8 27 𝑎1 𝑟3 = 8 27 ----------(1) Also 𝑎7 = − 64 729 𝑎1 𝑟6 = − 64 729 ----------(2) Dividing eq (1) from (2), we get 𝑎1 𝑟6 𝑎1 𝑟3 = − 644 729 8 27 𝑟3 = − 64 729 × 27 8 𝑟3 = − ( 2 3 ) 3 , real root only. 𝑟 = − 2 3 Put 𝑟 = − 2 3 in eq (1) 𝑎1 [− 2 3 ] 3 = 8 27 − 8 27 𝑎1 = 8 27 𝑎1 = −1 𝑎 𝑛 = 𝑎1 𝑟 𝑛−1 𝑎 𝑛 = (−1)[− 2 3 ] 𝑛−1 𝑎 𝑛 = (−1) 𝑛 [ 2 3 ] 𝑛−1
- 5. Geometric Mean Let aand b be two numbers. A number G is said to be geometric mean between two numbers a and b if a, G, b are in G.P, so 𝐺 𝑎 = 𝑏 𝐺 𝐺2 = 𝑎𝑏 𝐺 = ±𝑎𝑏 Example: Q: Find G.M between 2i and -4i Sol: Here 𝑎 = 2𝑖, 𝑏 = −4𝑖 𝐺 = ±√ 𝑎𝑏 𝐺 = ±√(2𝑖)(−4𝑖) 𝐺 = ±√−8𝑖2 𝐺 = ±2√2 Harmonic Progression If the reciprocal of the terms of the sequence form an A.P, then sequence is called harmonic. Example: Q: Find the 8th term of H.P: 1 2 , 1 5 , 1 8 , … Sol: Since 1 2 , 1 5 , 1 8 , … are in H.P
- 6. Therefore 2,5,8,… arein A.P 𝑎1 = 2, 𝑑 = 2 𝑎8 = 𝑎1 + 7𝑑 𝑎8 = 2 + 7(3) 𝑎8 = 2 + 21 𝑎8 = 23 In H.P 𝑎8 = 1 23 Harmonic Mean Let a and b be two numbers, then a number H is called Harmonic mean between two numbers a and b if a,H,b are in H.P Example 𝐻 = 2𝑎𝑏 𝑎 + 𝑏 16 = 2𝑎(4) 𝑎 + 4 16( 𝑎 + 4) = 8𝑎 16𝑎 + 64 = 8𝑎 16𝑎 − 8𝑎 = −64 8𝑎 = −64 𝑎 = −8
- 7. Uses of Sequence A sequence is a discrete structure used to represent an ordered list. A sequence is a function from a subset of the set of integers (usually either the set {0,1,2.. ..} Or {1,2, 3,...}To a set S. We use the notation an to denote the image of the integer n. ... Notation to represent sequence is {an} Advantages: The common 2-year sequence works well for many disciplines. Topics can be introduced "just- in-time" for many disciplines. Since all students take the same sequence, advising is relatively easy. Difference Between Sequences and Series Sequences Series Set of elements that follow a pattern Sum of elements of the sequence Order of elements is important Order of elements is not so important Finite sequence: 1,2,3,4,5 Finite series: 1+2+3+4+5 Infinite sequence: 1,2,3,4,…… Infinite Series: 1+2+3+4+……