1
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Question

# If a,a1,a2,.....,a2n,b are in arithmetic progression and a,g1,g2,....g2n,b are in geometric progression and h is the harmonic mean of a and b, then a1+a2ng1g2n+a2+a2nâˆ’1g2g2nâˆ’1+.....+an+an+1gngn+1 is equal to

A
2nh
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B
n/h
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C
nh
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D
2n/h
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Solution

## The correct option is D 2n/hSince a,a1,a2,....a2n,b are in arithmetic progressiona,a1,a2,....a2n are A.M's between a and b∴a1+a2n=a2+a2n−1=....=a+bsimilarly a,g1,g2,....g2n are in G.P∴g1g2n=g2g2n−1=....=gngn+1=ab∴a1+a2ng1g2n+a2+a2n−1g2g2n−1+....an+an+1gngn+1=a+bab+a+bab+......+a+bab=n(a+b)ab=2nhwhere h is H.M between a and b⇒h=2aba+b

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