Sequences and Series 5 Question 8

8. Let a and b be positive real numbers. If a,A1,A2,b are in arithmetic progression, a,G1,G2,b are in geometric progression and a,H1,H2,B are in harmonic progression, then show that

G1G2H1H2=A1+A2H1+H2=(2a+b)(a+2b)9ab

(2002, 5M)

Show Answer

Solution:

  1. Since, a,A1,A2,b are in AP.

A1+A2=a+ba,G1,G2,b are in GP G1G2=ab

and a,H1,H2,b are in HP.

H1=3ab2b+a,H2=3abb+2a1H1+1H2=1a+1bH1+H2H1H2=A1+A2G1G2=1a+1b

Now,

G1G2H1H2=ab3ab2b+a3abb+2a=(2a+b)(a+2b)9ab

From Eqs. (i) and (ii), we get

G1G2H1H2=A1+A2H1+H2=(2a+b)(a+2b)9ab



जेईई के लिए मॉक टेस्ट

एनसीईआरटी अध्याय वीडियो समाधान

दोहरा फलक