Applied Mathematics - Sequence and Series

Name: Applied Mathematics - Sequence and Series
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Last updated 4/2022

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What you'll learn

Introduction
Sequences
Series
Arithmetic Progression (A.P.)
Geometric Progression (G. P.)
Relationship Between A.M. and G.M.
Sum to n Terms of Special Series

Course content

2 sections • 33 lectures • 3h 36m total length

Introduction6:40
Explore sequences and series through real-world examples to reveal ordered number patterns and practical notation. Distinguish finite and infinite sequences, and learn terms like B1 and B2.
Arithmetic Progression5:56
Discover arithmetic progression with a constant difference, as in 2, 5, 8, 11, 14. Use the formula a_n = a + (n-1)d to find terms.
Arithmetic Progression - Sum of n Terms9:33
Derive the sum of the first n terms of an arithmetic progression using formulas: S_n = n/2 (a_1 + a_n) and S_n = n/2 (2a_1 + (n-1)d), illustrated with 2,5,8,11,14.
Types of Means (Arithmetic mean)3:25
Explore arithmetic mean in sequences and series. Learn that the mean of two numbers x and y is (x+y)/2, and for any number of terms it's (x1+x2+...+xm)/m.
Geometric Progressions5:42
Identify the common ratio in geometric progressions, derive the nth term formula ar^{n-1}, and explore real-world examples like doubling bacteria and salary increments.
Geometric Progressions (Sum of n Terms)7:56
Explore how to sum the first n terms of a geometric progression using the standard formula, including cases r ≠ 1 and the special case r = 1.
Geometric Progressions (Sum of Infinite Terms)5:47
This lecture shows how to sum a geometric progression to infinity. When |r|<1, the sum is a/(1-r); if not, the series diverges, as shown by a=1, r=1/2 giving 2.
Geometric Progressions (Property of Sum of n terms)5:22
Explore the property S_n minus S_{n-1} equals the nth term in a geometric progression, and verify it with the example 3, 6, 12, 24 and r = 2.
Types of Means (Geometric Mean)3:37
Learn the geometric mean, defined as the square root of the product of numbers with the same sign, and extended to multiple numbers by taking the root of their product.
Expressing Recurring Decimals as Rational Numbers4:45
Express recurring decimals as rational numbers by treating them as a geometric series and using the sum a/(1 - r); e.g., 0.4 recurring equals 4/9, and 3.4 recurring equals 31/9.
Arithmetico-Geometric Progressions (AGP)6:26
Explore arithmetical progression and geometric progression, define arithmetico-geometric progression as the product of corresponding terms, and derive its nth-term formula with examples.
Sum of n Terms of AGP10:00
We derive the sum of the first n terms of an arithmetico-geometric progression by forming S_n, subtracting, and simplifying to the main formula with (1−r) and (1−r)^2 in the denominator.
Sum to Infinity of AGP5:42
Discover how to compute the sum to infinity of an arithmetic-geometric progression using the r less than 1 condition, with an example sequence 1, 2/4, 3/16, 4/64, yielding 16/9.
Harmonic Progressions4:48
Learn how harmonic progressions relate to arithmetic progressions through reciprocals, derive end terms of hps using ap techniques, and work with examples like 1/2, 1/5, 1/8, 1/11.
Types of Means (Harmonic mean)4:03
Explore harmonic mean concept using reciprocals; for two numbers, HM = 2xy/(x+y), and for n numbers, HM = n divided by the sum of reciprocals, with examples.
Relation Between am, gm and hm6:50
Explore the relation among arithmetic mean, geometric mean, and harmonic mean, with formulas am = (x+y)/2, gm = sqrt(xy), hm = 2xy/(x+y).
am, gm and hm (If x = y)3:12
Explore the three classic means—arithmetic, geometric, and harmonic—and show that when two numbers are equal, all three means coincide at the common value.
Sum of First n Natural Numbers5:14
Discover the formula for the sum of the first n natural numbers, derived by pairing terms to yield n(n+1)/2, with a 1 to 20 example equal to 210.
Sum of Squares of First n Natural Numbers8:53
discover the formula for the sum of squares of the first n natural numbers and verify it with 1^2+2^2+3^2+4^2+5^2=55, expressed as n(n+1)(2n+1)/6.
Sum of Cubes of First n Natural Numbers12:48
Derive the sum of cubes of the first ten natural numbers using the sum of cubes formula, and verify it alongside the sum of squares.

Level - 1 Question -12:45
Compute the sum from r = 1 to n of 3r^2 - 2r + 1 by separating constants and using standard sum formulas, yielding (2n^2 + n + 1)/2.
Level - 1 Question -23:46
Explore evaluating a sum of cubes using the identity 1^3+...+n^3 = [n(n+1)/2]^2 and apply standard sums of squares and natural numbers to arrive at the result.
Level - 1 Question -32:29
Apply the difference of squares to the alternating sum of squares from 70 to 1, reduce each pair to (a+b)(a-b), and use the triangular number formula n(n+1)/2 to sum 1–70.
Level - 1 Question -45:50
Explore level-1 questions on sequences, linking geometric progression and arithmetic progression by examining end-term products, reciprocal sums, and derived formulas for sums and terms.
Level - 1 Question -55:09
Apply the geometric series sum formula to express S_n, S_{2n-1}, and S_{3n}, then verify the derived identity through factoring and aligning terms.
Level - 1 Question -61:19
Apply the harmonic mean formula to two positive numbers, with a = 15/2 and g = x, to solve for x. Conclude x = 24/5 using the harmonic mean calculation.
Level - 1 Question -71:53
To make a harmonic progression, insert 1/7 and 1/10 between 1/4 and 1/13. The reciprocals form an arithmetic progression with difference 3.
Level - 1 Question -84:26
Determine two numbers whose arithmetic mean exceeds the geometric mean by 2 and the harmonic mean by 18/5, giving 4 and 16.
Level - 1 Question -93:34
Learn to use the sums of first n natural numbers, squares, and cubes—n(n+1)/2, n(n+1)(2n+1)/6, and (n(n+1)/2)^2—by substitution to solve for unknowns.
Level - 1 Question -103:20
Explore level 1 geometric series problems in the sequence and series module, applying the sum formula S_n = a(r^n−1)/(r−1) with r = 2 to compute S7 and related terms.
Level - 1 Question -115:30
Learn to solve a sequence and series problem by converting an eight-filled sequence into a nine-based geometric progression, applying GP sum formulas to find the total.
Level - 1 Question -124:40
Identify a geometric progression from the sums of finite terms and of their squares equals 15, and derive the first term and ratio to obtain the GP 15/4, 15/16, 15/64.
Level - 2 Solved Questions - 1045:17
explore solving geometric progression problems, including finding GP terms from product and ratio conditions, summing finite and infinite GP, and applying GP to compound interest and distance problems.

Requirements

Basic knowledge of mathematics of 9th and 10th std Mathematics

Description

Sequence and Series

Sequence and Series
Arithmetic Progression (A.P.)
Arithmetic Mean (A.M.)
Geometric Progression (G.P.)
General term of a G.P.
Sum of n terms of a G.P.
Arithmetic and Geometric series infinite G.P. and its sum
Geometric mean (G.M.)
Relation between A.M. and G.M.

SUMMARY

1. By a sequence, we mean an arrangement of number in definite order according to some rule. Also, we define a sequence as a function whose domain is the set of natural numbers or some subsets of the type {1, 2, 3, ....k}. A sequence containing a finite number of terms is called a finite sequence. A sequence is called infinite if it is not a finite sequence.

2. Let a1 , a2 , a3 , ... be the sequence, then the sum expressed as a1 + a2 + a3 + ... is called series. A series is called finite series if it has got finite number of terms. ®

3. An arithmetic progression (A.P.) is a sequence in which terms increase or decrease regularly by the same constant. This constant is called common difference of the A.P. Usually, we denote the first term of A.P. by a, the common difference by d and the last term by l. The general term or the n th term of the A.P. is given by an = a + (n – 1) d. The sum Sn of the first n terms of an A.P. is given by

Sn = n/2 [2a + (n-1)d ] = n/2 (a + 1).

4. The arithmetic mean A of any two numbers a and b is given by (a + b) / 2 i.e., the sequence a, A, b is in A.P.

5. A sequence is said to be a geometric progression or G.P., if the ratio of any term to its preceding term is same throughout. This constant factor is called the common ratio. Usually, we denote the first term of a G.P. by a and its common ratio by r.

6. The geometric mean (G.M.) of any two positive numbers a and b is given by the sequence a, G, b is G.P.

Who this course is for:

Complete Mathematics for Engineering Entrance Exam Preparation. ( IIT-JEE Main | Advanced | BITSAT | SAT | etc.)
Those preparing for board and competitive exams State Board, CBSE, ICSE , IGCSE, MHT-CET & NEET
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Applied Mathematics - Sequence and Series

What you'll learn

Explore related topics

Course content

Sequence and Series20 lectures • 2hr 7min

Numericals (Level 1 | Level 2 | Level 3)13 lectures • 1hr 30min

Requirements

Description

Who this course is for: