NIMCET 2022 — Mathematics PYQ

1. Analyze the Series:

The given series is $\frac{\sqrt{5}}{3},\frac{\sqrt{5}}{4},\frac{1}{\sqrt{5}},\dots$

Notice that $\frac{1}{\sqrt{5}}$ can be written as $\frac{\sqrt{5}}{5}$ by rationalizing or simply multiplying the numerator and denominator by $\sqrt{5}$.

So, the series is: $\frac{\sqrt{5}}{3},\frac{\sqrt{5}}{4},\frac{\sqrt{5}}{5},\dots$

2. Check for Arithmetic Progression (AP) in Reciprocals:

A series is in Harmonic Progression (HP) if the reciprocals of its terms form an Arithmetic Progression (AP).

Let's take the reciprocals of the terms:

This sequence is an AP because the common difference ($d$) is constant:

3. Use the AP General Term Formula:

The reciprocal of the target term $\frac{\sqrt{5}}{13}$ is $\frac{13}{\sqrt{5}}$. Let this be the $n^{th}$ term of the AP.

The formula for the $n^{th}$ term of an AP is:

Substitute the values:

• $a_n=\frac{13}{\sqrt{5}}$

• $a=\frac{3}{\sqrt{5}}$

• $d=\frac{1}{\sqrt{5}}$

4. Solve for $n$:

Multiply the entire equation by $\sqrt{5}$ to simplify:

Conclusion:

The $11^{th}$ term of the given series is $\frac{\sqrt{5}}{13}$.

Correct Option:

B) 11