Question

1 + 3 + $5^2$ + 7 + $9^2$ + $\ldots$ upto 40 terms is equal to

• A. 43890
• B. 41880
• C. 33980
• D. 40870

Hint:

Separate the series into parts and sum each part individually.

Answer 1

The Correct Option is B

1. Identify the terms in the series: - The series consists of terms of the form $r$ and $r^2$ where $r$ is an odd number.
2. Separate the series into two parts: - Part 1: Sum of terms of the form $r$. - Part 2: Sum of terms of the form $r^2$.
3. Sum of terms of the form $r$: - The sequence is $1, 3, 5, 7, \ldots$ up to 20 terms. - Sum of the first 20 odd numbers: \[ \sum_{r=1}^{20} (2r-1) = 20^2 = 400 \]
4. Sum of terms of the form $r^2$: - The sequence is $1^2, 3^2, 5^2, 7^2, \ldots$ up to 20 terms. - Sum of the squares of the first 20 odd numbers: \[ \sum_{r=1}^{20} (2r-1)^2 = \sum_{r=1}^{20} (4r^2 - 4r + 1) \] \[ = 4 \sum_{r=1}^{20} r^2 - 4 \sum_{r=1}^{20} r + \sum_{r=1}^{20} 1 \] \[ = 4 \cdot \frac{20 \cdot 21 \cdot 41}{6} - 4 \cdot \frac{20 \cdot 21}{2} + 20 \] \[ = 4 \cdot 2870 - 4 \cdot 210 + 20 = 11480 - 840 + 20 = 10660 \] 5. Total sum of the series: \[ \text{Total sum} = 400 + 10660 = 41880 \] Therefore, the correct answer is (2) 41880.