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Question

The sum of first four terms of a geometric progression (G.P.) is 6512 and the sum of their respective reciprocals is 6518. If the product of first three terms of the G.P. is 1, and the third term is α, then 2α is 


Solution

Let the first four terms be a,ar,ar2,ar3.
a+ar+ar2+ar3=6512     (1) 
and 1a+1ar+1ar2+1ar3=6518
1a(r3+r2+r+1r3)=6518     (2) 
(1)(2), we get
a2r3=1812=32

Also, a3r3=1
a(32)=1a=23
49r3=32
r3=3323r=23
α=ar2=23.(32)2=32
2α=3

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