Question

The value of $a{C_0} + (a + b){C_1} + (a + 2b){C_2} + ....... + (a + nb){C_n}$ Is equal to
A. $(2a + nb){2^n}$
B. $(2a + nb){2^{n - 1}}$
C. $(na + 2b){2^n}$
D. $(na + 2b){2^{n - 1}}$

Answer 1

Hint : In these types of questions the students are advised to understand the combination theory.
This type of questions involve the series combination involving the coefficients ranging till the n terms.
Students are advised to go through the series expansion theory to attempt such type of questions.
This is an example of a binomial theorem.
Binomial theorem used in this example because it is used to expand the equation with powers.
{(1 = x)^n} = \sum\limits_{k = 0}^n {\left( {\begin{array}{*{20}{c}} n \\\ k \end{array}} \right)} {x^k}
According to the theorem, it is possible to expand the polynomial ${(x + y)^n}$ into a sum involving terms of the form $a{x^b}{y^c}$ ,
where the exponents b and c are nonnegative integers with $b + c = n$ , and the coefficients of each term is a specific positiveinteger depending on $n\,and\,b.$

Complete step-by-step answer :
From the given question,
We have the equation,
$a{C_0} + (a + b){C_1} + (a + 2b){C_2} + ....... + (a + nb){C_n}$
Where,
${C_{0,}}{C_{1,}}........,{C_N}$ denote the binomial coefficients.
Now, let's get back to the equation,
$a{C_0} + (a + b){C_1} + (a + 2b){C_2} + ....... + (a + nb){C_n}$
According to Binomial Theorem we know that,
$\sum\limits_{r = 0}^n {(a + rb){}^n} {C_r}$ is true for some r
The calculations follow as below
We will expand the above form because
$a{C_0} + (a + b){C_1} + (a + 2b){C_2} + ....... + (a + nb){C_n}$ is equal to $\sum\limits_{r = 0}^n {(a + rb){}^n} {C_r}$
Thus, we get the following derivation by step by step approaching the problem.
$\sum\limits_{r = 0}^n {a{}^n} {C_r} + \sum\limits_{r = 0}^n {rb{}^n} {C_r} \\\ \Rightarrow a(\sum\limits_{r = 0}^n {{}^n} {C_r}) + {b^n}(\sum\limits_{r = 0}^n {r{}^n} {C_r}) \\\ \Rightarrow a(\sum\limits_{r = 0}^n {{}^n} {C_r}) + {b^n}(\sum\limits_{r = 0}^n {r.\dfrac{n}{r}{}^{n - 1}} {C_{r - 1}}) \\\ \Rightarrow a\sum\limits_{r = 0}^n {{}^n} {C_r} + {b^n}\sum\limits_{r = 0}^n {{}^{n - 1}} {C_{r - 1}} \\\ \Rightarrow a{2^n} + bn{2^{n - 1}} \\\ \Rightarrow (2a + nb){2^{n - 1}}$

So, the correct answer is “Option B”.

Note : Students might make mistakes while dealing with the operations that involve use of summation and combination.
Students will understand the use of the terms $r{\text{ }}and{\text{ }}C$ after revising the Binomial theorem.
Students must understand what is asked in such a question and should be able to classify the data correctly.
Students might mess up while putting the values in the actual equation.