JEE Mathematics 2023 Question with Solution

Given: The series is

$$2.2^2 - 3^2 + 2.4^2 - 5^2 + 2.6^2 - 7^2 + \cdots$$

Find: The sum to $20$ terms.

From the solution, the series is written as

$$S = \sum_{n=1}^{10} (2n + 0.2)^2 - (2n+1)^2$$

Expanding,

$$(2n + 0.2)^2 = 4n^2 + 0.8n + 0.04$$

and

$$(2n+1)^2 = 4n^2 + 4n + 1$$

Therefore,

$$(2n + 0.2)^2 - (2n+1)^2 = -3.2n - 0.96$$

Now summing from $n=1$ to $10$,

$$\sum_{n=1}^{10} (-3.2n) = -3.2 \times \frac{10(11)}{2} = -176$$

and

$$\sum_{n=1}^{10} (-0.96) = -9.6$$

Hence,

$$S = -176 - 9.6 = -185.6$$

However, the solution finally states that the correct sum is $1310$. Following the solution's stated conclusion, the answer is $1310$. There is a clear discrepancy between the shown working and the stated final answer.

Given: The source solution contains intermediate algebra and a final stated answer.

Find: The answer consistent with the solution's.

The intermediate working shown in the working gives

$$S = \sum_{n=1}^{10} \left[(2n+0.2)^2 - (2n+1)^2\right] = -185.6$$

But the same source then concludes: "Therefore, the correct sum to the series is $1310$".

Since the extraction rule gives primary source to the solution conclusion, the recorded answer is $1310$. Students should note that the displayed derivation and the final stated answer on the page do not agree.