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NIMCET 2008 — Mathematics PYQ

NIMCET | Mathematics | 2008

If H1,H2,…,Hn are n harmonic means between a and b, $a \neq= b$ , then the value of $H 1 - a H 1 + a + H n - b H n + b$ is equal to:

Choose the correct answer:

Explanation

Understand Harmonic Means: If H1,…,Hn are HM between a,b, then a1,H11,…,Hn1,b1 are in AP.

Let common difference be d:

$H 1 1 = a 1 + d ⟹ H 1 1 - a 1 = d ⟹ a H 1 a - H 1 = d$ .

$b 1 = H n 1 + d ⟹ b 1 - H n 1 = d ⟹ b H n H n - b = d$ .

Evaluate terms: * $H 1 - a H 1 + a = 1/ a - 1/ H 1 1/ a + 1/ H 1 = - d 2/ a + d = - a d 2 - 1$ .

$H n - b H n + b = 1/ b - 1/ H n 1/ b + 1/ H n = d 2/ b - d = b d 2 - 1$ .

Sum: Using the AP property $b 1 = a 1 + (n + 1) d$ , the sum simplifies to 2n.

Answer: (c) 2n

Choose the correct answer:

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