JAMIA 2026 — Mathematics PYQ

Step 1: Identify the components of the Arithmetic Progression

• First term ($a$): $1$

• Common difference ($d$): $1$

• Last term ($l$): $\frac{n-2}{2}$

Step 2: Find the total number of terms ($N$)

Since the series starts at $1$ and increases consecutively by $1$ up to the last term, the total number of terms $N$ is simply equal to the value of the last term itself:

$$N=\frac{n-2}{2}$$

Step 3: Calculate the sum of the series ($S_N$)

The formula for the sum of an Arithmetic Progression when the first term ($a$) and the last term ($l$) are known is:

$$S_N=\frac{N}{2}\cdot [a+l]$$

Now, let's substitute our values into this formula:

$$S_N=\frac{\left(\frac{n-2}{2}\right)}{2}\cdot \left[1+\frac{n-2}{2}\right]$$

Simplify the fraction outside the bracket:

$$S_N=\frac{n-2}{4}\cdot \left[1+\frac{n-2}{2}\right]$$

Now, simplify the terms inside the bracket by taking a common denominator of $2$:

$$1+\frac{n-2}{2}=\frac{2+(n-2)}{2}=\frac{n}{2}$$

Substitute this simplified value back into our sum equation:

$$S_N=\frac{n-2}{4}\cdot \frac{n}{2}$$

$$S_N=\frac{n\times (n-2)}{8}$$

Conclusion

The sum of the given series is $\frac{n\times (n-2)}{8}$.

Therefore, the correct option is (a).