Question

The $20\text{th}$term of the series $2\times 4+4\times 6+6\times 8+....$ will be:

• A. $1600$
• B. $1680$
• C. $420$
• D. $840$

Answer 1

The correct option is B

$1680$

Step 1: Finding the general term of the sequence.

Given series is $2\times 4+4\times 6+6\times 8+....$

Writing the series in general term, we get

$$\begin{gathered} 2\times 4+4\times 6+6\times 8+.... \\ \Rightarrow \left[2\times 1(2\times 1+2)\right]+\left[2\times 2(2+2+2\right]+....\left[2n(2n+1)\right] \\ \therefore T_n=\left[2n(2n+1\right]\dots \dots \left(1\right) \end{gathered}$$

The series is in arithmetic progression where the consecutive terms differ by a constant difference.

Step 2: Calculating the $20\text{th}$ term.

Substituting $n=20$ in equation $\left(1\right)$ we get

$$\begin{gathered} T_{20}=\left[2\times 20(2\times 20+2\right] \\ =1680 \end{gathered}$$

Therefore, the $20\text{th}$ term is $1680$

Hence, the correct answer is Option (B).