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

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.