Let a,b,c>0 and a,b,c are in A.P. a^2,b^2,c^2 are in G.P. then choose the correct one ? (a)a=b=c, (b)a^2+b^2=c^2 , (c) a^2+c^2=3 b^2, (d) none of these

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Answer 1 · 2017-09-28T11:30:33

#a=b=c#

The generic terms of an AP sequence can be represented by:

# sf({a, a+d, a+2d }) #

We are told that #{a,b,c}#, and we note that if we take a higher term and subtract its previous term we get the common difference; thus

# c-b = b-a #
# :. 2b = a+c # ..... [A]

The generic terms of an GP sequence can be represented by:

# sf({a, ar, ar^2 }) #

We are told that #{a^2, b^2, c^2}#, and we note that if we take a higher term and divide by its previous term we get the common ratio, thus:

# c^2/b^2 = b^2/a^2 => c/b = b/a \ \ \ \ # (as #a,b,c gt 0 #)
# :. b^2 = ac # ..... [B]

Substituting [A] into [B] we have:

# ((a+c)/2)^2 = ac #
# :. a^2 + 2ac + c^2 = 4ac #
# :. a^2 - 2ac + c^2 = 0 #
# :. (a-c)^2 = 0 #
# :. a=c #

And if we substitute #a=c# into Eq [B], we have:

# b^2 = c^2 => b= c \ \ \ \ # (as #a,b,c gt 0 #)

Hence we have #a=c# and #b=c => a=b=c#