Question

The \[{20^{{\rm{th }}}}\] term of the series \[2 \times 4 + 4 \times 6 + 6 \times 8 + \ldots \ldots \] will be
A. 1600
B. 1680
C. 420
D. 840

Answer 1

Hint:
To answer this question, we must first get the total number of terms by using the formula \[{a_n} = 2n \times (2n + 2)\] then determine the 20th term from the end by using the formula. Then we have to substitute the term asked for and proceed with solving the equation and thus obtained the required solution.
Formula use:
\[{a_n} = 2n \times (2n + 2)\]
Complete step-by-step solution
We have been given the series in the question,
\[2 \times 4 + 4 \times 6 + 6 \times 8 + \ldots \ldots \]
To find the \[{(n)^{th}}\] of the series, we can use the below formula
\[{a_n} = 2n \times (2n + 2)\]
\[ = 4{n^2} + 4n\]
In the question, we are asked to find the\[{(20)^{th}}\] term, for that we have to substitute the term asked to find in the formula \[{a_n} = 4{n^2} + 4n\]:
Now, the equation becomes,
\[{a_{20}} = 4{(20)^2} + 4(20)\]
Calculate the exponents \[{(20)^2} = 400\]:
Replace the value in the above equation, we obtain
\[ = 4 \cdot \:400 + 4\left( {20} \right)\]
Now, we have to multiply and divide left to right, we obtain
\[ = 1600 + 4\left( {20} \right)\]
Multiply the number from the above equation, we get
\[{\rm{ = 1600 + 80}}\]
Let’s add the numbers in the above equation, we obtain
\[{\rm{ = 1680}}\]
Therefore, the \[{(20)^{th}}\]term is \[1680\]
Hence, the option B is correct.
Note:
When such questions arise, the most common error a person might do is to enter the incorrect value into the formula. Furthermore, another strategy can be used to answer this question. That is by considering the last term of the series and using that in the formula \[{a_n} = 2n \times (2n + 2)\]. Students are more prone to make mistakes if they do not comprehend the question's phrasing and calculate the sum of the first \[{(20)^{th}}\] of the series, resulting in an incorrect solution.