Arithmetic Mean, Geometric Mean, Harmonic Mean

Arithmetic Mean, Geometric Mean, Harmonic Mean and its relation

1. Arithmetic Mean, Geometric Mean & Harmonic Mean Dr. N. B. Vyas Department of Science & Humanities ATMIYA University

2. Arithmetic Mean If three 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 If three 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.

4. Q.1 Find A.M. & G.M. of following numbmers : 8 and 32 2 and 18 1/32 and 8

5. Q.2 The AM and GM of two numbers are 25.5 and 12 respectively, find the numbers.

6. Q.3 The AM of 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 three numbers 3, k+3 and 4k are in G.P. find the value of k.

8. Q.5 The sum of 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 person has 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 A series x 1 , x 2 , x 3 ,…., x n is said to be in Harmonic Progression when their resicprocals 1/x 1 , 1/x 2 , 1/x 3 , … , 1/ x n 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 When three 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 is no 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 the 29 th 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 T n = a + (n-1) d For n = 29 T 29 = _____ =88 Therefore, 1 / 88 is the 29 th 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 = GM 2

16. Ex Find HM of the following: ( i ) 2 and 32 (ii) 8 and 18 (iii) ½ and 8

17. Ex For two numbers 5 and 44, verify that ( i ) G 2 = A. H (ii) H < G < A

18. Ex A person pays 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 difference between two consecutive installments is Rs. 5 i.e. constant, hence the installments form an AP First installment = Rs. 100 a =100, d = - 5 , S n = 975