In a geometric progression in which all the terms are positive, the third term exceeds the first term by 32 while the fifth term exceeds the first term by 320. Find the common ratio and the first term. ?

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Answer 1 · 2018-07-09T12:50:33

#:. a=4, r=3#.

We consider the GP :

#{a,ar,ar^2,ar^3,ar^4,...}={ar^(n-1) | n in NN}#.

By what is given,

#ar^2=a+32......(1), and, ar^4=a+320.......(2)#.

#:. ar^2-a=32......(1'), and, ar^4-a=320......(2')#.

#:. (2') -: (1') rArr {a(r^4-1)}/{a(r^2-1)}=320/32=10, i.e., #,

# r^2+1=10 rArr r^2=9 rArr r=+-3#.

# r=+-3, and, (1) rArr 9a=a+32 rArr a=4#.

#r=-3, &, a=4," give the GP"={4,-12,36,...}#, which is

impossible, as, it is given that all the terms of GP are #+ve#.

#:. a=4, r=3#.