Answer

Verified

381.3k+ views

**Hint:**In the given question we need to use the concept of binomial expansion and then also make use of the definition of combination as in the question we are been asked the sum of the coefficients of the binomial expansion of the given term ${{\left( 1+x \right)}^{n}}$. And the sum of coefficients involves n+1 coefficients of the given expansion.

**Complete step-by-step solution:**

According to the given question, we need to find the sum of the coefficients of the n+1 terms of the expansion. Also, we need to find the sum of ${{C}_{0}}+\dfrac{{{C}_{1}}}{2}+\dfrac{{{C}_{2}}}{3}+...+\dfrac{{{C}_{n}}}{n+1}$.

So, we can also write this as $\sum\limits_{r=0}^{r=n}{\dfrac{{{C}_{r}}}{r+1}}$ .

Now, we know that the $r^{th}$ coefficient of binomial expansion is given by ${}^{n}C_{r}=\dfrac{n!}{r!\left( n-r \right)!}$ .

Now, using the above two expression we get:

$\sum\limits_{r=0}^{r=n}{\dfrac{{{C}_{r}}}{r+1}}=\dfrac{n!}{r!\left( n-r \right)!}\times \dfrac{1}{r+1}$

Now, multiplying and dividing by n+1 the above term we get, $\begin{align}

& \sum\limits_{r=0}^{r=n}{\dfrac{{{C}_{r}}}{r+1}}=\dfrac{\left( n+1 \right)n!}{r!\left( n-r \right)!}\times \dfrac{1}{r+1}\times \dfrac{1}{\left( n+1 \right)} \\

&= \dfrac{1}{\left( n+1 \right)}\sum\limits_{r=0}^{r=n}{\dfrac{\left( n+1 \right)!}{\left( r+1 \right)!}\times \dfrac{1}{\left( n-r \right)!}} \\

&= \dfrac{1}{\left( n+1 \right)}\sum\limits_{r=0}^{r=n}{\dfrac{\left( n+1 \right)!}{\left( r+1 \right)!}\times \dfrac{1}{\left( n+1-\left( r+1 \right) \right)!}} \\

&= \dfrac{1}{\left( n+1 \right)}\sum\limits_{r=0}^{r=n+1}{C_{r+1}^{n+1}} \\

\end{align}$

Now, we know that the expansion of ${{\left( 1+x \right)}^{n}}$is ${{\left( 1+x \right)}^{n}}=1+nx+\dfrac{n\left( n-1 \right)}{2!}{{x}^{2}}+...$

Now, when we take x=1, we get ${}^{n}C_{0}+{}^{n}C_{1}+....+{}^{n}C_{r}+...$

Now similarly if we take n+1 term we get, ${}^{n+1}C_{0}+{}^{n+1}C_{1}+....+{}^{n+1}C_{n+1}$ , now using the above expression we get

$\begin{align}

&= \dfrac{1}{\left( n+1 \right)}\sum\limits_{r=0}^{r=n}{{}^{n+1}C_{r+1}}-\dfrac{1}{\left( n+1 \right)}{}^{n+1}C_{0}+\dfrac{1}{\left( n+1 \right)}{}^{n+1}C_{0} \\

&= \dfrac{1}{\left( n+1 \right)}\sum\limits_{r=0}^{r=n}{{}^{n+1}C_{r}}-\dfrac{1}{\left( n+1 \right)}{}^{n+1}C_{0} \\

& = \dfrac{{{2}^{n+1}}-1}{n+1} \\

\end{align}$

**Therefore, the sum of the coefficients of the binomial expansion of the term ${{\left( 1+x \right)}^{n}}$is $\dfrac{{{2}^{n+1}}-1}{n+1}$.**

**Note:**In these types of questions, we need to take care of the calculation initially. Now, we need to be very careful with the concept of the combination, then only we will be able to find out the way in which we can get the sum of the given expansion. Such questions are although easy to solve but we get confused in the concept itself as these questions are not that direct.

Recently Updated Pages

what is the correct chronological order of the following class 10 social science CBSE

Which of the following was not the actual cause for class 10 social science CBSE

Which of the following statements is not correct A class 10 social science CBSE

Which of the following leaders was not present in the class 10 social science CBSE

Garampani Sanctuary is located at A Diphu Assam B Gangtok class 10 social science CBSE

Which one of the following places is not covered by class 10 social science CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

The states of India which do not have an International class 10 social science CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

How do you graph the function fx 4x class 9 maths CBSE

One Metric ton is equal to kg A 10000 B 1000 C 100 class 11 physics CBSE

Difference Between Plant Cell and Animal Cell

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Why is there a time difference of about 5 hours between class 10 social science CBSE

Name the three parallel ranges of the Himalayas Describe class 9 social science CBSE