Let A be the sum of the first 20 terms and B be the sum of the first 40 terms of the series 1² + 2.2² + 3² + 2.4² + 5² + 2.6² + .....
If B – 2A = 100λ, then λis equal to

• 496

• 232

• 248

• 464

C.

248

$$\begin{gathered} A =1^2 + 2.2^2 + 3^2+...2.{20}^2 \\ = (1^2 + 2^2 + 3^2+ .....{20}^2) + 4(1^2 +2^2 + 3^2 + ......+ {10}^2) \\ = \frac{20 x 21x41}{6} + \frac{4 x 10 x 11x 21}{6} \\ = 2870 + 1540 = 4410 \\ B = 1^2 + 2.2^2 + 3^2+...2.{40}^2 \\ = (1^2 + 2^2 + 3^2+ ....{40}^2) + 4(1^2 +2^2 + 3^2 + ......+ {20}^2) \\ = \frac{40 x 41 x 81}{6} + \frac{4 x 20 x 21 x 41}{6} \\ = 22140 + 11480 = 33620 \\ \Rightarrow \hspace{0.17em}B- 2A = 33620 - 8820 = 24800 \\ \Rightarrow 100\lambda = 24800 \\ \lambda = 248 \end{gathered}$$