Question

Find the value of :- ${C_1} + 2{C_2} + 3{C_3} + ... + n{C_n}$
A.${2^{n - 1}}$
B.${2^{n + 1}}$
C.$n{2^{n - 1}}$
D.$n{2^{n + 1}}$

Answer 1

Hint: We will use the formula for the expansion of a binomial term and find the binomial expansion of ${\left( {1 + x} \right)^n}$. After simplifying it, we will differentiate both sides of the equation with respect to $x$. We will use the formula for the differentiation of ${x^n}$ with respect to $x$ to do so. After differentiating the equation, we will substitute 1 for $x$ in the new equation that we have obtained. After substituting 1 in the new equation, we will obtain the given expression on the right-hand side of the equation and the required answer on the left-hand side of the equation. We will choose the correct option accordingly.
Formulas used:
We will use the following formulas:
1.The binomial expansion of ${\left( {a + b} \right)^n}$ is given by ${\left( {a + b} \right)^n} = {}^n{C_0}{a^n}{b^0} + {}^n{C_1}{a^{n - 1}}{b^1} + ... + {}^n{C_n}{a^0}{b^n}$.
2.The differentiation of ${x^n}$ with respect to $x$ is given by $\dfrac{d}{{dx}}\left( {{x^n}} \right) = n{x^{n - 1}}$.

Complete step-by-step answer:
We will calculate the binomial expansion of ${\left( {1 + x} \right)^n}$ by substituting 1 for $a$ and $x$ for $b$ in the formula ${\left( {a + b} \right)^n} = {}^n{C_0}{a^n}{b^0} + {}^n{C_1}{a^{n - 1}}{b^1} + ... + {}^n{C_n}{a^0}{b^n}$. Therefore, we get
\[{\left( {1 + x} \right)^n} = {}^n{C_0}{1^n}{x^0} + {}^n{C_1}{1^{n - 1}}{x^1} + {}^n{C_2}{1^{n - 1}}{x^2}... + {}^n{C_n}{1^0}{x^n}\]
Simplifying the above equation, we get
\[ \Rightarrow {\left( {1 + x} \right)^n} = {}^n{C_0} + {}^n{C_1}{x^1} + {}^n{C_2}{x^2}... + {}^n{C_n}{x^n}\]…………………………………\[\left( 1 \right)\]
We will differentiate both sides of equation \[\left( 1 \right)\] with respect to $x$.
\[ \Rightarrow \dfrac{d}{{dx}}{\left( {1 + x} \right)^n} = \dfrac{d}{{dx}}\left( {{}^n{C_0} + {}^n{C_1}{x^1} + {}^n{C_2}{x^2}... + {}^n{C_n}{x^n}} \right){\text{ }}\]
Using the formula $\dfrac{d}{{dx}}\left( {{x^n}} \right) = n{x^{n - 1}}$$x$ in the above equation, we get
\[ \Rightarrow n{\left( {1 + x} \right)^{n - 1}} = 0 + {}^n{C_1} + 2{}^n{C_2}x + ... + n{x^{n - 1}}\]
Substituting 1 for $x$ in the above equation, we get
\[ \Rightarrow n{\left( {1 + 1} \right)^{n - 1}} = 0 + {}^n{C_1} + 2{}^n{C_2} \cdot 1 + ... + n \cdot 1 \\
   \Rightarrow n{2^{n - 1}} = {}^n{C_1} + 2{}^n{C_2} + 3{}^n{C_3} + ... + n \\ \]
The expression on the right-hand side of the equation is the required expression.
So, the value of ${C_1} + 2{C_2} + 3{C_3} + ... + n{C_n}$ is $n{2^{n - 1}}$.
$\therefore $ Option C is the correct option.

Note: The binomial theorem is not only used to expand expressions of higher powers, but also used to find probability and to solve complex problems of economics, algebra, calculus etc. It is also used in designing infrastructure.
We have used the concept of differentiation to simplify the equation further. Differentiate helps to find the derivative of a function with respect to any variable. It measures the rate of change in an independent variable.