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.
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 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 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
• 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 29th 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 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) G2 = 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 , Sn = 975