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![[Post New]](/templates/default/images/icon_minipost_new.gif) 6 Dec 2007 12:35:38 IST
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if a,b,c,d are in H.P. then prove that ab+bc+cd=3da
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6 Dec 2007 12:46:18 IST
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hiii
ma ans is cmng 3ad
see
abcd are in HP
abc are in HP , bcd are in HP
b = 2ac / a + c
c = 2bd/b+d
a+c = 2ac/b
b+d = 2bd/c
(a+c)(b+d) = (2ac/b)(2bd/c)
ab + ad + bc + cd = 4ad
ab + bc + cd = 3ad
hence its the ans
~~~~~~~~~~~~ALREADY DONE IN OTHER TOPIC~~~~~~~~~~~
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6 Dec 2007 12:47:39 IST
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1/a, 1/b, 1/c, 1/d r in ap
1/b - 1/a = 1/c-1/b = 1/d-1/c=(1/d-1/a)/3
or (a-b)/ab = (b-c)/bc=(c-d)/cd=(a-d)/3ad = k(say)
a-b = kab, b-c=kbc,c-d=kcd,a-d=3kad
(a-b)+(b-c)+(c-d)-(a-d) = k(ab+bc+cd-3ad) =0
so ab+bc+cd=3ad
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