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Arithmetic Mean, Geometric Mean, Harmonic Mean
This document discusses different types of means - arithmetic mean, geometric mean, and harmonic mean. It provides definitions and formulas for calculating each type of mean between two numbers. It also presents examples of calculating means and solving word problems involving arithmetic progressions and geometric/harmonic progressions. The key information covered includes definitions of arithmetic, geometric, and harmonic means; formulas for calculating each mean; and examples of applying the concepts to word problems.
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Overview of Arithmetic Mean, Geometric Mean, and Harmonic Mean presented by Dr. N. B. Vyas.
Definition of AM with examples. The formula A = (a+b)/2 and practice questions Q1 on calculating AM. Definition of GM with examples and the formula for GM. Includes practice question to find numbers based on AM and GM. Practice questions on calculating AM & GM and solving equations involving their relationships.
Questions related to finding values in G.P. and sums of numbers in A.P. converted to G.P.
Problem involving payments in A.P. for a debt of Rs. 19600 with final installment calculations.
Definition of H.P. and H.M. with examples. Relationship between terms and A.P. characteristics.
Conversion of a series in H.P. to A.P. and finding specific terms in H.P. with an example.
The relationship between AM, GM, and HM is explored with examples and verification exercises.
Calculation problem of monthly installments paid in A.P., detailing first installment and series properties.
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Arithmetic Mean, Geometric Mean, Harmonic Mean
- 1. Arithmetic Mean, Geometric Mean& Harmonic Mean Dr. N. B. Vyas Department of Science & Humanities ATMIYA University
- 2. Arithmetic Mean • Ifthree numbers are in A.P. then the middle number is said to be the Arithmetic Mean (AM) of the first and the third numbers. • E.g. ▫ 3,5,7 are in A.P. then 5 is A.M. of 3 & 7 ▫ 10, 16, 22 are in A.P. then 16 is A.M. of 10 & 22 • If a and b are two numbers and if their A.M. is denoted by A then a, A, b are in A.P. • A= (a+b) / 2
- 3. Geometric Mean • Ifthree number are in G.P. then the middle number is said to be Geometric Mean(G.M.) of the first and third numbers. • E.g. ▫ 1, 6, 36 are in GP then 6 is GM of 1 & 36 ▫ 5, 10, 20 are in GP then 10 is GM of 5 & 20 • If a and b are two numbers and their G.M. is denoted by G the a, G, b are in G.P. abGabG 2
- 4. Q.1 • Find A.M.& G.M. of following numbmers: 1. 8 and 32 2. 2 and 18 3. 1/32 and 8
- 5. Q.2 • The AMand GM of two numbers are 25.5 and 12 respectively, find the numbers.
- 6. Q.3 • The AMof two numbers exceeds their positive GM by 10 and the first number is 9 times the second number, find the two numbers.
- 7. Q.4 • If threenumbers 3, k+3 and 4k are in G.P. find the value of k.
- 8. Q.5 • The sumof three numbers in AP is 30. If 2, 4 and 3 are deducted from them respectively the resulting form G.P. • Find the numbers
- 9. Q.6 • A personhas to pay a debt of Rs.19600 in 40 monthly installments, which are in A.P. But after paying 30 installments he dies, leaving Rs. 7,900 unpaid. • Find the first installment paid by him.
- 10. Harmonic Progression • Aseries x1, x2, x3,….,xn is said to be in Harmonic Progression when their resicprocals 1/x1 , 1/x2 , 1/x3 , … , 1/xn are in Arithmetic progression. • Eg: • ½ , ¼, 1/6 , 1/8, … • 1/5 , 1/8 , 1/11 , 1/14 , … • are in Harmonic Progression • “if a, b, c are in H.P. then 1/a , 1/b , 1/c are in AP”
- 11. Harmonic Mean • Whenthree numbers are in H.P., the middle number is called the Harmonic Mean between the other two numbers. • If a, H, b are in H.P. then H is the Harmonic Mean of a and b. • Also 1/a, 1/H, 1/b are in A.P. • H = 2ab / (a+b)
- 12. Note: • There isno general formula for the sum of any number of terms in HP. • Generally first we convert the given series into AP and then use the properties of AP.
- 13. Ex • Find the29th term of the series • ¼, 1/7, 1/11, 1/14,….
- 14. Solution: • ¼, 1/7,1/11, 1/14,…. are in HP • Therefore, 4, 7, 11, 14, …. are in AP • Here a=4, d=3 • Tn = a + (n-1) d • For n = 29 • T29 = _____ • =88 • Therefore, 1/88 is the 29th term of HP.
- 15. Relation between AM,GM and HM • For any two real numbers • HM ≤ GM ≤ AM • AM . HM • = {(a+b) / 2 } . { 2ab / (a+b)} • = ab • = GM2
- 16. Ex • Find HMof the following: • (i) 2 and 32 • (ii) 8 and 18 • (iii) ½ and 8
- 17. Ex • For twonumbers 5 and 44, verify that • (i) G2 = A. H • (ii) H < G < A
- 18. Ex • A personpays Rs.975 by monthly installments each less than the former by Rs. 5. • The first installment is Rs. 100 • In what time entire amount be paid?
- 19. Solution: • There differencebetween two consecutive installments is Rs. 5 i.e. constant, hence the installments form an AP • First installment = Rs. 100 • a =100, d = - 5 , Sn = 975