Question

The coefficient of $x$ in the expansion of $\left( {1 + x} \right)\left( {1 + 2x}
\right)\left( {1 + 3x} \right).....\left( {1 + 100x} \right)$ is –
(A) \[5050\]
(B) \[10100\]
(C) \[5151\]
(D) \[4950\]

Answer 1

Hint: Series – An expression of the form ${x_1} + {x_2} + {x_3}......,$where ${x_1},{x_2},{x_3}......,$ is a sequence of numbers is called a series. Coefficient is a numerical or constant quantity placed before and multiplying the variable in an algebraic expression.

Complete step by step answer:
$\left( {1 + x} \right)\left( {1 + 2x} \right)\left( {1 + 3x} \right).....\left( {1 + 100x} \right)$
We are required to find the coefficient of $x$ in the above expansion.
Now, first, multiply the first two terms
$\left( {1 + x} \right)\left( {1 + 2x} \right) = 1 + x + 2x + 2{x^2}$
$\Rightarrow 1 + x\left( {1 + 2} \right) + 2{x^2}$
Similarly, multiply the first three terms
$\Rightarrow \left( {1 + x} \right)\left( {1 + 2x} \right)\left( {1 + 3x} \right) \\
\Rightarrow \left( {1 + x + 2x + 2{x^2}} \right)\left( {1 + 3x} \right){\text{ }}\left[ {{\text{from previous multiplicaiton we can write}}\left( {1 + x} \right)\left( {1 + 2x} \right)} \right] \\
\Rightarrow 1 + x + 2x + 2{x^2} + 3x + 3{x^2} + 6{x^2} + 6{x^3}\left[ { = \left( {1 + x + 2x + 2{x^2}} \right)} \right] \\
\Rightarrow 1 + \left( {x + 2x + 3x} \right) + 2{x^2} + 3{x^2} + 6{x^2} + 6{x^3} \\
\Rightarrow 1 + x\left( {1 + 2 + 3} \right) + 11{x^2} + 6{x^3} \\$
So, in the expansion of $\left( {1 + x} \right)\left( {1 + 2x} \right)\left( {1 + 3x} \right).....\left( {1 + 100x} \right)$ coefficient of $x$ should be $\left( {1 + 2 + 3 + ..... + 100} \right)$. We know that the sum of the first $'n'$ natural numbers is given by the expression $\dfrac{{n\left( {n + 1} \right)}}{2}$.
$\therefore $ The coefficient of $x$ in the above expression is $ = \dfrac{{100\left( {100 + 1} \right)}}{2}$
$= \dfrac{{100 \times 101}}{2} \\
= 5050 \\$

$\therefore $ The correct answer is option \[\left( A \right){\text{ }}5050.\]

Note: Here students must take care of the concept of series. Students made a mistake during the multiplication. They think that in a series there are so many terms, how will we multiply them. So, they should be aware about a series and their properties.