NIMCET 2021 — Mathematics PYQ

If $a,H_1,H_2,\dots ,H_n,b$ are in Harmonic Progression (HP), then their reciprocals are in Arithmetic Progression (AP):

Let $d$ be the common difference of this AP.

The term $\frac{1}{b}$ is the $(n+2{)}^{th}$ term:

Step 1: Finding $H_1$

The second term of the AP is $\frac{1}{H_1}$:

Taking the reciprocal and rearranging for the required form:

To get $\frac{H_1+a}{H_1-a}$, we use the property of ratios or simplify directly:

Step 2: Finding $H_n$

The $(n+1{)}^{th}$ term of the AP is $\frac{1}{H_n}$:

Similarly:

Step 3: Adding the two expressions

Since we know $(n+1)d=\frac{1}{b}-\frac{1}{a}$, we substitute $\left(\frac{1}{b}-\frac{1}{a}\right)=(n+1)d$:

Final Answer: