Question

Given in the expansion of \[{{\left( 1+x \right)}^{43}},\] the coefficient of \[{{\left( 2r+1 \right)}^{th}}\] and coefficient of \[{{\left( r+2 \right)}^{th}}\] terms are equal. Find the value of r.

Answer 1

Hint: To solve the given question, we will first find out what binomial expansion of any term \[{{\left( a+b \right)}^{m}}\] is. Then we will try to write the general term of this expansion in terms of m, a, and b. After writing the general term, we will replace m with 43, a with 1 and b with x. Now, we will determine the coefficient of this general term. With the help of this, we will find the coefficient of \[{{\left( r+2 \right)}^{th}}\] term and \[{{\left( 2r+1 \right)}^{th}}\] term. Then, we will form two cases. In case I, we will equate the coefficients of \[{{\left( r+2 \right)}^{th}}\] term and \[{{\left( 2r+1 \right)}^{th}}\] term. From here, we will get a value of r. In case II, we will make use of the condition that if \[{{\left( i+1 \right)}^{th}}\] and \[{{\left( j+1 \right)}^{th}}\] term of \[{{\left( a+b \right)}^{m}}\] are equal, then i + j = m. From here, we will get another value of r.

Complete step-by-step answer:
To start with, let us first find out what binomial expansion is and what will be the binomial expansion of \[{{\left( a+b \right)}^{m}}.\] The binomial expansion describes the algebraic expansion of powers of a binomial. The binomial expansion of \[{{\left( a+b \right)}^{m}}\] is given as shown below.
\[{{\left( a+b \right)}^{m}}={{\text{ }}^{m}}{{C}_{0}}{{a}^{m}}{{b}^{0}}+{{\text{ }}^{m}}{{C}_{1}}{{a}^{m-1}}{{b}^{1}}+{{\text{ }}^{m}}{{C}_{2}}{{a}^{m-2}}{{b}^{2}}+{{\text{ }}^{m}}{{C}_{3}}{{a}^{m-3}}{{b}^{3}}+........+{{\text{ }}^{m}}{{C}_{m-1}}{{a}^{1}}{{b}^{m-1}}+{{\text{ }}^{m}}{{C}_{m}}{{a}^{0}}{{b}^{m}}\]
Here, we can see that the first term is \[^{m}{{C}_{0}}{{a}^{m}}{{b}^{0}},\] the second term is \[^{m}{{C}_{1}}{{a}^{m-1}}b\] and so on. Thus, we can say that the general term of this expansion would be
\[\text{General term or }{{\text{p}}^{th}}\text{ term}={{\text{ }}^{m}}{{C}_{p-1}}{{a}^{m-p+1}}{{b}^{p-1}}\]
Thus, the general term of \[{{\left( 1+x \right)}^{43}}\] will be equal to
\[\text{General term or }{{\text{p}}^{th}}\text{ term}={{\text{ }}^{43}}{{C}_{p-1}}{{\left( 1 \right)}^{43-p+1}}{{\left( x \right)}^{p-1}}\]
\[\Rightarrow \text{General term or }{{\text{p}}^{th}}\text{ term}={{\text{ }}^{43}}{{C}_{p-1}}\times 1\times {{x}^{p-1}}\]
Now, the coefficient of this general term is \[^{43}{{C}_{p-1}}.\] Thus, we can say that the \[{{\left( 2r+1 \right)}^{th}}\] term’s coefficient will be \[^{43}{{C}_{2r+1-1}}={{\text{ }}^{43}}{{C}_{2r}}.\]
Similarly, \[{{\left( r+1 \right)}^{th}}\] term’s coefficient will be \[^{43}{{C}_{r+2-1}}={{\text{ }}^{43}}{{C}_{r+1}}.\]
Now, we are given that these coefficients are equal. Thus, we have two cases.
Case 1: As both the coefficients are equal, we have,
\[^{43}{{C}_{2r}}={{\text{ }}^{43}}{{C}_{r+1}}\]
Thus, 2r = r + 1
Case 2: If \[^{m}{{C}_{a}}={{\text{ }}^{m}}{{C}_{b}}\] then we can say that a + b = m. Thus, we will get,
\[\Rightarrow 2r+r+1=43\]
\[\Rightarrow 3r+1=43\]
\[\Rightarrow 3r=42\]
\[\Rightarrow r=14\]
Thus, the values of r are 1 and 14.

Note: We can also write the binomial expansion of \[{{\left( a+b \right)}^{m}}\] as shown below.
\[{{\left( a+b \right)}^{m}}={{\text{ }}^{m}}{{C}_{0}}{{a}^{0}}{{b}^{m}}+{{\text{ }}^{m}}{{C}_{1}}{{a}^{1}}{{b}^{m-1}}+{{\text{ }}^{m}}{{C}_{2}}{{a}^{2}}{{b}^{m-2}}+\text{ }........+{{\text{ }}^{m}}{{C}_{m}}{{a}^{m}}{{b}^{0}}\]
It does not depend on how we write the expansion of \[{{\left( a+b \right)}^{m}},\] the answer will be the same because the coefficients of the expansion of \[{{\left( a+b \right)}^{m}}\] have the property \[^{m}{{C}_{p}}={{\text{ }}^{m}}{{C}_{m-p}}.\]