AI Mentor

Check Your IQ

Free Expert Demo

Try Test

Courses

Dropper NEET Course Dropper JEE Course Class - 12 NEET Course Class - 12 JEE Course Class - 11 NEET Course Class - 11 JEE Course Class - 10 Foundation NEET Course Class - 10 Foundation JEE Course Class - 10 CBSE Course Class - 9 Foundation NEET Course Class - 9 Foundation JEE Course Class -9 CBSE Course Class - 8 CBSE Course Class - 7 CBSE Course Class - 6 CBSE Course

Offline Centres

If A is the arithmetic mean and G1, G2be two geometric means between any two numbers, then $\frac{{(G_{1})}^{2}}{G_{2}} + \frac{{(G_{2})}^{2}}{G_{1}} = 2 A .$ State whether the statement is true or false.

see full answer

Your Exam Success, Personally Taken Care Of

1:1 expert mentors customize learning to your strength and weaknesses – so you score higher in school , IIT JEE and NEET entrance exams.

An Intiative by Sri Chaitanya

True

False

answer is A.

(Unlock A.I Detailed Solution for FREE)

Best Courses for You

JEE

NEET

Foundation JEE

Foundation NEET

CBSE

Detailed Solution

Let us assume a and b are the two numbers and ‘r' be the common ratio.
We are given that A is the arithmetic mean of a and b which means

A = \frac{a + b}{2}

2A= a+b -------------------------------------(1)

G1,G2 are two geometric means between a and b.
a,G1,G2,b are in Geometric progression.
if ‘a’ the first term and ‘b’ is the fourth term in the G.P. and nth term of a geometric progression is

T_{n}

{ar}^{n - 1}

‘b’ is the fourth term i.e. n=4

T_{4}

{ar}^{4 - 1}

{ar}^{3}

\frac{b}{a} = r^{3}

\sqrt[3]{\frac{b}{a}} = r

G_{1}

is the second term i.e. n=2

T_{2}

{ar}^{2 - 1} = ar

G_{1}

=ar=a

\sqrt[3]{\frac{b}{a}}

G_{2}

is the third term i.e. n=3

T_{3}

{ar}^{3 - 1}

G_{2}

{ar}^{2} = a . {(\sqrt[3]{\frac{b}{a}})}^{2}

Substitute the values of

{G_{1}, G}_{2}, \frac{{(G_{1})}^{2}}{G_{2}} + \frac{{(G_{2})}^{2}}{G_{1}}

\frac{{(ar)}^{2}}{{ar}^{2}} + \frac{{({ar}^{2})}^{2}}{ar} = \frac{{a^{2} r}^{2}}{{ar}^{2}} + \frac{{a^{2} {(r}^{2})}^{2}}{ar} = a + a r^{3}

But

\sqrt[3]{\frac{b}{a}} = r

\frac{{(G_{1})}^{2}}{G_{2}} + \frac{{(G_{2})}^{2}}{G_{1}} = a + a {(\sqrt[3]{\frac{b}{a}})}^{3} = a + a . \frac{b}{a} = a + b

from equation (1) we get

\frac{{(G_{1})}^{2}}{G_{2}} + \frac{{(G_{2})}^{2}}{G_{1}} = 2 A

Watch 3-min video & get full concept clarity

courses

No courses found

Result Producing online coaching for JEE/NEET of south India

Largest All India Test Series for JEE/NEET

Best Online Course for IIT JEE

Best Online Course for NEET

Best Online Course for CBSE

Ready to Test Your Skills?

Check your Performance Today with our Free Mock Test used by Toppers!

Take Free Test

Get Expert Academic Guidance – Connect with a Counselor Today!