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Question
If b is the mean proportional between a and c, prove that (ab + bc) is the mean proportional between (a² + b²) and (b² + c²).
Solution
b is the mean proportional between a and c then
b2 = ac …(i)
Now if (ab + bc) is the mean proportional
between (a2 + b2) and (b2 + c2), then
(ab + bc)2 = (a2 + b2) (b2 + c2)
Now L.H.S. = (ab + bc)2
= a2b + b2c2 + 2ab2c
= a2(ac) + ac(c)2 + 2a ac.c ...[from (i)]
= ac(a2 + c2 + 2ac)
= ac(a + c)2
R.H.S. = (a2 + b2)(b2 + c2)
= (a2 + ac)(ac + c2) ...[from (i)]
= a(a + c) c(a + c)
= ac(a + c)2
∴ L.H.S. = R.H.S.
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