If a, b, c are in HP, then:

Solution

Maths sir · 3 steps

In an HP, flip every term: the reciprocals form an AP, and the AP rule gives you b.

1. Flip to an AP. a, b, c are in HP exactly when 1/a, 1/b, 1/c are in AP. In an AP the middle term is the average of its neighbours, so 2/b = 1/a + 1/c.

   2/b = 1/a + 1/c = (a + c)/ac

2. Solve for b. Take the reciprocal of both sides: b/2 = ac/(a+c), so b = 2ac/(a+c). This is the harmonic mean of a and c.
3. So. The relation is b = 2ac/(a+c), which is option (a).

The idea to keep. An HP is an AP in disguise: take reciprocals, use the AP rule, then flip back. The middle term of an HP is 2ac/(a+c).

Answer from the NTA final answer key. Solution by PrepWiser.