Join now for JEE/NEET and also prepare for Boards Join now for JEE/NEET and also prepare for Boards. Register Now
Hey there! We receieved your request
Stay Tuned as we are going to contact you within 1 Hour
One of our academic counsellors will contact you within 1 working day.
Click to Chat
1800-1023-196
+91-120-4616500
CART 0
Use Coupon: CART20 and get 20% off on all online Study Material
Welcome User
OR
LOGIN
Complete Your Registration (Step 2 of 2 )
Sit and relax as our customer representative will contact you within 1 business day
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.
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.
Signing up with Facebook allows you to connect with friends and classmates already using askIItians. It’s an easier way as well. “Relax, we won’t flood your facebook news feed!”
Dear , Preparing for entrance exams? Register yourself for the free demo class from askiitians.
Harmonic Mean Table of Content Formula for...
Examples Based on Relationship between AM, GM and...
Arithmetic Mean between two numbers When three...
Geometric Progression (G.P.) Table of contents...
Relationship between A.M. and G.M. Table of...
Arithmetic Progression (A.P.) Table of contents...
Sum of first n terms of AP A sequence of numbers...
Arithmetic Mean Examples What exactly do you mean...
Weighted Means Let a 1 , a 2 , ……, a...
Basic Principle of AP Progressions and series...
Solved Examples based on GP Solved examples :...
Sum to n terms of Special Series Table of contents...
Solved Examples Based on Harmonic mean...
Sequences Table of contents Meaning of Sequence...
Geometric Progression Geometric Progression as the...
Method of Differences Suppose a 1 , a 2 , a 3 ,...
Download IIT JEE Solved Examples Based on...
Basic Concepts of Sequences and Pattern Table of...
Introduction of Sequences and Series Table of...
Harmonic Progression Harmonic progression is an...
Arithmeticogeometric Progression Solved Examples...
Arithmetic-Geometric Progression Suppose a 1 , a 2...
Geometric Mean Geometric Mean between two numbers...
Arithmetic Mean of mth Power The concept of...
Series Table of contents Meaning of Series History...