JEE 2023 — Mathematics PYQ

1. Find the General Term ($T_r$)

Looking at the pattern, the $r^{th}$ term of the series can be written as:

2. Factorize the Denominator

We can simplify the denominator using the identity for Sophie Germain's factorization:

Using the difference of squares $a^2-b^2=(a-b)(a+b)$:

3. Decompose into Partial Fractions

Now, rewrite $T_r$ so that the numerator $r$ is expressed in terms of the factors in the denominator:

4. Find the Sum ($S_{10}$)

The sum of the first 10 terms is $S_{10}={\sum }_{r=1}^{10}{T}_r$. This is a telescoping series:

• For $r=1$: $T_1=\frac{1}{2}\left(\frac{1}{1}-\frac{1}{3}\right)$

• For $r=2$: $T_2=\frac{1}{2}\left(\frac{1}{3}-\frac{1}{7}\right)$

• For $r=3$: $T_3=\frac{1}{2}\left(\frac{1}{7}-\frac{1}{13}\right)$

• ...

• For $r=10$: $T_{10}=\frac{1}{2}\left(\frac{1}{10^2-10+1}-\frac{1}{10^2+10+1}\right)=\frac{1}{2}\left(\frac{1}{91}-\frac{1}{111}\right)$

Summing them up, all middle terms cancel out:

Correct Option: (1)