Thank you for registering.

One of our academic counsellors will contact you within 1 working day.

Please check your email for login details.
Close

MY CART (5)

Use Coupon: CART20 and get 20% off on all online Study Material

ITEM
DETAILS
MRP
DISCOUNT
FINAL PRICE
Total Price: Rs.

⇐ BackProceed to Checkout ⇒
There are no items in this cart.
Continue Shopping

Relation between A.M., G.M. and H.M.


Let there are two numbers ‘a’ and ‘b’, a, b > 0

then AM = a+b/2

 GM =√ab

 HM =2ab/a+b

 ∴ AM × HM =a+b/2 × 2ab/a+b = ab = (√ab)2 = (GM)2

 Note that these means are in G.P.

 Hence AM.GM.HM follows the rules of G.P.

 i.e. G.M. =√A.M. × H.M.

 Now, let us see the difference between AM and GM

 AM – GM =a+b/2 – √ab

=(√a2)+(√b)–2√a√b/2

 i.e. AM > GM

 Similarly,

 G.M. – H.M. = √ab –2ab/a+b

=√ab/a+b (√a – √b)2 > 0

 So. GM > HM

 Combining both results, we get

 AM > GM > HM                                                   …….. (12)

 All sequences of numbers cannot be put into A.P./G.P./H.P. Let us study these.
 

Important Points

r3 (r – 1)3 = 3 r2 – 3r + 1

r = 1 : 13 – 0 = 3 . 12 – 3 . 1 + 1

r = 2 : 23 – 13 = 3 . 22 – 3 . 2 + 1

r = 3 : 33 – 23 = 3 . 32 – 3 . 3 + 1

r = n : n3 – (n–1)3 = 3.(n2) – 3(n) + 1

Adding

n3 = 3 (12 + 22 +…+ n2) –3 (1 + 2 + 3 +…+ n) + (1 + 1 +…+ n times)

n3 = 3 Σnr=1 r2 – 3 (n(n+1))/2 + n

⇒ 3 Σnr=1 r2 = n3 + 3n(n+1)/2 – n

= n/2 (2n2 + 3n + 1)

⇒ Σnr=1 r2 = n(n+1)(2n+1)/6

To read more, Buy study materials of Sequences and Series comprising study notes, revision notes, video lectures, previous year solved questions etc. Also browse for more study materials on Mathematics here.

Algebra

OFFERED PRICE: Rs. 2,332
View Details Xpress Buy

Course Features

  • 101 Video Lectures
  • Revision Notes
  • Test paper with Video Solution
  • Mind Map
  • Study Planner
  • NCERT Solutions
  • Discussion Forum
  • Previous Year Papers



Course Features

  • 731 Video Lectures
  • Revision Notes
  • Previous Year Papers
  • Mind Map
  • Study Planner
  • NCERT Solutions
  • Discussion Forum
  • Test paper with Video Solution