NCERT Solutions
Class 11 Maths
Sequences and Series
Ex.9.3 Q.29
If A and G be A.M. and G.M., respectively between two positive numbers, prove that the numbers are A ± √ {(A + G) (A - G)}
It is given that A and G are A.M. and G.M. between two positive numbers.
Let these two positive numbers be a and b.
Now, AM = A = (a + b) ÷2 …………... (1)
and
GM = G = √ab …………. (2)
From equation (1) and (2), we obtain
a + b = 2A ................... (3)
ab = G2 ....................... (4)
Substituting the value of a and b from 3 and 4 in the identity (a – b)2 = (a + b)2 – 4ab, we get
(a – b)2 = 4A2 – 4 G2
(a – b)2 = 4(A2 – G2)
(a – b)2 = 4(A – G) (A + G)
(a – b) = 2√ {(A + G) (A – G)} ………... (5)
From equation (3) and (5), we get
2a = 2A + 2√ {(A + G) (A – G)}
=> a = A + √ {(A + G) (A – G)}
Substituting the value of a in (3), we obtain
a = 2A - A - √ {(A + G) (A – G)}
=> b = A - √ {(A + G) (A – G)}
Thus, the two numbers are A ± √ {(A + G) (A - G)}