Target Audience: High School Students, College Freshmen and Sophomores, Class 11/12 Students in India preparing for ISC/CBSE and Entrance Examinations like the IIT-JEE Main or Advanced/AIEEE, and anyone else who needs this Tutorial as a reference! (Tutorial on AP, GP, HP, AM, GM, HM and Series Summations)- With Solved Problems, MCQ Quizzes Series and Progressions : AP, GP, HP After studying the chapter you might find it useful to attempt these Multiple Choice Question Quizzes to assess how well you understood the topic. A sequence refers to a list of numbers with some inherent pattern. General term of a sequence, generally written as a_{n} is an expression that for different values of ‘n’ gives different terms of the sequence. A Quick Overview: There is a detailed tutorial document further down, but here's a quick glance at the basic concepts Simple Examples: Computing the Arithmetic Mean Arithmetic Progression, Terms and SummationA sequence of the type a,a+d,a+2d,a+3d,… is said to form an AP. Each term differs from the previous one by a fixed value called as “common difference”. a and d are usually used to denote first term and common difference respectively. nth term of an AP is given by a_{n}=a_{1}+(n-1)d The tutorial will show you how to compute the sum of the first n terms of an APSelecting terms in AP:3 terms in an AP are: a-d, a, a+d 4 terms in an AP are: a-3d, a-d, a+d, a+3d And so on. m^{th} term from the end of an AP : It is the (n-m+1)^{th} term from the beginning. Alternately, we can reverse the AP, and consider the first term and common difference of the new AP as last term and negative of common difference of the old AP. If a constant is added, or subtracted, or multiplied or divided from every term on an AP, the resulting sequence is still an AP. Arithmetic Mean:if a,b,c are in AP, then b is said to be arithmetic mean of a and c. Equivalently, if 2b=a+c, then a,b,c are in AP. If between a and b we are to insert n numbers A1,A2,A3,….,An such that a, A1,A2,A3,….,An form an AP, then A1,A2,A3,….,An are called n Arithmetic Means. Example 1: Compute the Arithmetic mean of -5, -15, 12, 5, -4, 12, -15, 1, 2, -14 Arithmetic mean = (Sum of elements)/(number of elements) = (-5 + -15 + 12 + 5 + -4 + 12 + -15 + 1 + 2 + -14)/2 = -21/2 = -10.5 Example 2: Compute the Arithmetic mean of -11, 4, -14, 2, 3, -3, 1, -8, -5, -6 Arithmetic mean = (Sum of elements)/(number of elements) = (-11 + 4 + -14 + 2 + 3 + -3 + 1 + -8 + -5 + -6)/3 = -37/3 = -12.333 Example 3: Compute the Arithmetic mean of 11, 8, -3, 11, 4, 12, 13, -2, -6, 10 Arithmetic mean = (Sum of elements)/(number of elements) = (11 + 8 + -3 + 11 + 4 + 12 + 13 + -2 + -6 + 10)/4 = 58/4 = 14.5 Example 4: Compute the Arithmetic mean of 0, 7, 8, -15, 10, -12, -9, -7, 7, -5 Arithmetic mean = (Sum of elements)/(number of elements) = (0 + 7 + 8 + -15 + 10 + -12 + -9 + -7 + 7 + -5)/5 = -16/5 = -3.2 Geometric Progression, Terms and SummationA sequence of the type: a, ar, ar^{2}, ar^{3},… is called a GP. In a GP, the nth term of a GP is given by: an=ar^{m-1} The tutorial will show you how to compute the sum of first n terms of a GP. When r=1, the GP can be treated as an AP, and clearly the sum of first n terms then is “nXa”. Similar to an AP, m^{th} term from the end is (n-m+1)^{th} term from the starting. Example 1: Compute the Geometric mean of 8, 10, -8, 8, -15, 12, -12, 4, -7, 10<br/> Geometric mean = (Product of elements)/(number of elements) = (8 x 10 x -8 x 8 x -15 x 12 x -12 x 4 x -7 x 10)/2 = 3096576000/2 = 1548288000.0 Example 2: Compute the Geometric mean of -4, 2, 12, -13, -3, -1, -14, -5, 1, 2<br/> Geometric mean = (Product of elements)/(number of elements) = (-4 x 2 x 12 x -13 x -3 x -1 x -14 x -5 x 1 x 2)/3 = 524160/3 = 174720.0 Selecting terms in a GP:A GP of 3 terms : a/r, a, ar A GP of 4 terms: a/r^{3}, a/r, ar, ar^{3} And so on. If all terms in a GP are multiplied or divided by the same number, or are raised to the same power, then the resulting sequence is still a GP. Geometric Mean:If a, b, c are in GP then b^{2}=ac and b is called the GM of a and c. Conversely, if b^{2}=ac, then a,b,c are in GP. Sum of infinite terms of a GP: If -1<r<1, then GP is said to converge, that is to say that sum of infinite terms of such a GP tends to a constant value. The geometric mean of N given numbers is the Nth root of the product of those N numbers. For example: The geometric mean of 2, 4, 8 is the cubic root of 64 = 4 Harmonic ProgressionIf a sequence is in AP, then the sequence obtained by taking the reciprocal of every term in the sequence forms an HP. That is if a,b,c,… form an AP, then 1/a,1/b,1/c,… form an HP. Let a,b,c form an HP. Then clearly, 1/a,1/b,1/c form an AP. b is called the Harmonic Mean. Problems related to HP are generally solved by converting it into an AP. Example 1: Compute the Harmonic mean of 10, 10, 14, 2, 2, 5, -1, -11, 12, 11<br/> Geometric mean = (number of elements)/(Sum of reciprocal of elements) = (2/1/10 + 1/10 + 1/14 + 1/2 + 1/2 + 1/5 + 1/-1 + 1/-11 + 1/12 + 1/11)/ = 2/0.5547619047619046 = 3.605 Example 2: Compute the Harmonic mean of 1, -1, 13, -14, -1, -12, -1, -7, 14, -5<br/> Geometric mean = (number of elements)/(Sum of reciprocal of elements) = (5/1/1 + 1/-1 + 1/13 + 1/-14 + 1/-1 + 1/-12 + 1/-1 + 1/-7 + 1/14 + 1/-5)/ = 5/-2.349267399267399 = -2.128 Example 3: Compute the Harmonic mean of -15, 13, 9, -7, -12, -6, -1, -14, -6, 1<br/> Geometric mean = (number of elements)/(Sum of reciprocal of elements) = (6/1/-15 + 1/13 + 1/9 + 1/-7 + 1/-12 + 1/-6 + 1/-1 + 1/-14 + 1/-6 + 1/1)/ = 6/-0.5095848595848596 = -11.774 Relationship between Arithmetic, Harmonic and Geometric MeansNOTE: Let a,b be two positive real numbers. Then AM X HM=GM^{2}. Also, AM>GM>HM Here are some of the problems solved in the tutorial :Q: What is the common difference of an AP whose nth term is xn+y? Q: If the sum of n terms of an AP is 2n2+3n, what is the kth term? Q: If am=n and an=m, what is ap? Q: If the sums of n terms of two arithmetic progressions are in the ratio 2n+5:3n+4, then what is the ratio of their nth terms? Q: If sum of n terms of an AP is 3n2+5n then which of its terms is 164? Q: If sum of first p terms is q and sum of first q terms is p then what is the sum of first p+q terms ? Q: If four numbers are in AP such that their sum is 50, and the greatest number is four times the least, then what are the numbers? Q: If n arithmetic means are inserted between 1 and 31 such that ratio of first and nth mean is 3:29, then what is the value of n? Q: If S1 is the sum of an AP of ‘n’ odd number of terms and S2 the sum of terms of the series in odd places, then what is S1/S2? Q: What is the sum of 2 digit multiples of 4? Q: Find the sum of 2+6+18+…+4374. Q: Find the sum of 5+55+555+…n terms Q: Find the sum of n terms of the sequence whose nth term is 2n+3n Q: Find a GP for which the sum of first two terms is -4 and the fifth term is 4 times the third term. Q: Find the least value of n for which the sum of 1+3+9+27+… to n terms is greater than 7000. Q: If a,b,c are in GP, prove that log a, log b, log c are in AP and vice versa. Q: If a,b,c are in GP and x,y are AM’s between a,b and b,c respectively, then prove that x,b,y are in HP. Q: Find the sum to n terms of series 1.2.3+2.3.4+3.4.5+4.5.6+… Q: If a, b, c are in H.P., show that a/b+c, b/c+a, c/a+b are also in H.P. Q: Evaluate: 62 + 72 + 82 + 92 + 102 Q: Prove that (a + b + c) (ab + bc + ca) > 9abc. Q: If a2, b2, c2 are in A.P. show that b+ c, c, c+ a, a + b are in H.P. Q: If the AM between ‘a’ and ‘b’ is twice as greater as the GM, show that a/b = 7+4√3 Q: Sum to the n terms the series, 12-22+32-42+… n is even. Example of Standard Sums of first N terms of common sequencesThe Complete Tutorial with Solved Problems (MCQ Quizzes/Worksheets after this)MCQ Quiz |
Introduction to Complex Numbers Introduction to Complex Numbers and iota. Argand plane and iota. Complex numbers as free vectors. N-th roots of a complex number. Notes, formulas and solved problems related to these sub-topics. | The Principle of Mathematical Induction Introductory problems related to Mathematical Induction. | Quadratic Equations Introducing various techniques by which quadratic equations can be solved - factorization, direct formula. Relationship between roots of a quadratic equation. Cubic and higher order equations - relationship between roots and coefficients for these. Graphs and plots of quadratic equations. | Quadratic Inequalities Quadratic inequalities. Using factorization and visualization based methods. |
Series and Progressions Arithmetic, Geometric, Harmonic and mixed progressions. Notes, formulas and solved problems. Sum of the first N terms. Arithmetic, Geometric and Harmonic means and the relationship between them. |
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